First order differential equations with separable variables by Peter Stone, Dept. of Applied and Environmental Sciences, RMIT peter.stone@rmit.edu.au . . or . . peterstone@optusnet.com.au Version: 29.4.2003 restart; ; <Text-field layout="Heading 2" style="Heading 2">Introductory example </Text-field> ; Consider the differential equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEdGJiUieUdGKEYo . It can be solved by separating the variables and writing the equation in the form: NiMlInlH NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQlInhHRig= and then integrating both sides with respect to x : NiMvLSUkaW50RzYkKiYlInlHIiIiLSUhRzYjKiYlI2R5R0YpJSNkeEchIiJGKSUieEcsJC1GJTYkRjFGMUYw . This gives: NiMvLSUkaW50RzYkJSJ5R0YnLCQtRiU2JCUieEdGKyEiIg== , so NiMvKiYlInlHIiIjRiYhIiIsJiomJSJ4R0YmRiZGJ0YnJSJjRyIiIg== ,where NiMlImNH is an arbitrary constant, or NiMvKiQlInlHIiIjLCYqJCUieEdGJiEiIiYlImNHNiMiIiJGLg== , that is, NiMvLCYqJCUieEciIiMiIiIqJCUieUdGJ0YoJiUiY0c2I0Yo ------- (i), where NiMvJiUiY0c2IyIiIiomIiIjRidGJUYn .
Thus the solution curves are circles with their centre at the origin . Conversely, starting with the equation NiMvLCYqJCUieEciIiMiIiIqJCUieUdGJ0YoJiUiY0c2I0Yo , implicit differentiation shows that NiMsJiomIiIjIiIiJSJ4R0YmRiYqJkYlRiYlInlHRiZGJg== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh , which gives: NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEdGJiUieUdGKEYo .Solving the equation (i) for y gives: NiMvJSJ5Ry0lJXNxcnRHNiMsJiYlImNHNiMiIiJGLCokJSJ4RyIiIyEiIg== , or NiMvJSJ5RywkLSUlc3FydEc2IywmJiUiY0c2IyIiIkYtKiQlInhHIiIjISIiRjE= . Suppose that we are given the initial condition NiMvLSUieUc2IyIiISIiIg== . Substituting NiMvJSJ4RyIiIQ== and NiMvJSJ5RyIiIg== in equation (i) enables us to see that NiMvJiUiY0c2IyIiIkYn . Then we obtain the particular solution : NiMvJSJ5Ry0lJXNxcnRHNiMsJiIiIkYpKiQlInhHIiIjISIi . Note that the equation NiMvJSJ5RywkLSUlc3FydEc2IywmIiIiRioqJCUieEciIiMhIiJGLg== gives NiMvJSJ5RywkIiIiISIi when NiMvJSJ4RyIiIQ== , and so the initial condition is not satisfied. The graph of the solution NiMvJSJ5Ry0lJXNxcnRHNiMsJiIiIkYpKiQlInhHIiIjISIi is an arc of the unit circle. plot(sqrt(1-x^2),x=-1.1..1.1,numpoints=100,scaling=constrained); LSUlUExPVEc2LC0lJ0NVUlZFU0c2JDdgcTckJCEzUysrK0M+ciUqKiohIz0kIjNVOHNIbidwO0QkISM+NyQkITNkKioqKio0RUl4JCoqRiwkIjMwKGVuUiRSQjk2Riw3JCQhMyUpKioqKip6ZlsyKSkqRiwkIjMxKCpwVGREdVI6Riw3JCQhMz0rKysyKzR2KCpGLCQiM2pQNzw/cSQqM0BGLDckJCEzYysrKzs5VnAnKkYsJCIzWVQuKTQhMyMqXERGLDckJCEzJSkqKioqKkhyQzFYKkYsJCIzQWFEQGJ4KilvS0YsNyQkITNfKioqKioqM1hMQyMqRiwkIjMtKDNxayQ0Y2hRRiw3JCQhMyMpKioqKioqPit4KSoqKUYsJCIzLVJaZ0B3VmhWRiw3JCQhM18qKioqKnpAXG53KUYsJCIzKj1sbmVQSzMiW0YsNyQkITNvKioqKioqb15RaSYpRiwkIjNRKkdTNyRvIWU7JkYsNyQkITMoKioqKioqUmw+QkwpRiwkIjNkbik+KmVlQkhiRiw3JCQhM2wqKioqKnBZNDg1KUYsJCIzaylmMj9sKltpZUYsNyQkITM/KysrWiQpb3l5RiwkIjNxUjksLDNWZWhGLDckJCEzKSkqKioqKkgvRmxuKEYsJCIzJSlcKmZndWcnM2tGLDckJCEzWSsrKylwUGhWKEYsJCIzSy52Rk9VNCdvJ0YsNyQkITNgKioqKio0UidcS3NGLCQiMyIpKipcIXp4bWUhcEYsNyQkITMpKSoqKioqKlJKaCYqcEYsJCIzZ3QoeTV4RGQ5KEYsNyQkITN1KioqKipcS0pmeSdGLCQiM0UrJ2ZgcVZeTShGLDckJCEzVysrK1NyKGViJ0YsJCIzXTYwZClIIT5edkYsNyQkITNdKysrYTYib0wnRiwkIjNrV3lubE8lZnQoRiw3JCQhM0UrKyt5KFEjM2hGLCQiMyh6bWInKSpcbTx6Riw3JCQhM10qKioqKlwpekwpKmVGLCQiM3coNFxXb1lfMilGLDckJCEzXysrK0I7JD5uJkYsJCIzRnZ3bGl6JWVCKUYsNyQkITM3KysrViFmblYmRiwkIjNTJkgvbllfSFIpRiw3JCQhM00rKytAMS9LX0YsJCIzRSlveEpMdT9fKUYsNyQkITNgKysrOyJSNCwmRiwkIjNHTS96PCJIUmwpRiw3JCQhM3kqKioqKnBpPkR5JUYsJCIzR25ZejZuQSN5KUYsNyQkITMhKioqKioqNHNiIWZYRiwkIjM/S3ZpMzpHKyopRiw3JCQhM0QrKyssSCVHTSVGLCQiMzBaJ1wramV4KypGLDckJCEzKCoqKioqKnBVdkY1JUYsJCIzMFtUXzJyZz4iKkYsNyQkITMpKSoqKioqXDdrcSlRRiwkIjNiQi0hKSlvO09AKkYsNyQkITMxKysrbml1Y09GLCQiM2A9YiJRXUZ1SSpGLDckJCEzdSoqKioqZjBXIVtNRiwkIjNfbik9KmVidSdRKkYsNyQkITNGKysrOXUoKT5LRiwkIjMreUMoeVBRdVkqRiw3JCQhMy0rKysmUSE+MElGLCQiMyM0ZlE7O2V4YCpGLDckJCEzQysrKyJlIXkheSNGLCQiMydmTVEwIVtlMCcqRiw3JCQhM0ErKyt6Knk4YyNGLCQiMz0uYFdBRFNtJypGLDckJCEzISoqKioqKlIkZW9KQkYsJCIzTzNLWXlLT0MoKkYsNyQkITMpKSoqKioqZi5rLzYjRiwkIjNlO19hTS93dSgqRiw3JCQhMyMqKioqKipmektVKT1GLCQiM2o1XTZDInozIykqRiw3JCQhMy8rKytsWygpZjtGLCQiMyNmXlwsZnk3JykqRiw3JCQhMzMrKytHS3JgOUYsJCIzJz1ARydwO3gkKikqRiw3JCQhMy0rKytEKEd1QCJGLCQiM1RlLVBAbmhEKipGLDckJCEzJyoqKioqKipcSTQxNUYsJCIzRyIqXCM+NmcjXCoqRiw3JCQhMycqKioqKioqKlJedyF5Ri8kIjMpeldccXB0JXAqKkYsNyQkITMvKysrK2UoNGwmRi8kIjM9SkwpKm8vLSUpKipGLDckJCEzLSsrKytaRWNLRi8kIjNHQUEqUidwcCUqKipGLDckJCEzMysrKys0aSM9IkYvJCIzeXo0WHoxSSoqKipGLDckJCIzMSsrKytNKTQ8IkYvJCIzTzRXVnZWSioqKipGLDckJCIzOysrKyskUWdKJEYvJCIzM2hXdEsvXSUqKipGLDckJCIzISoqKioqKioqNCJSaWNGLyQiM10uZm8jemJSKSoqRiw3JCQiM2cqKioqKioqeicqb28oRi8kIjMxOk9UTz9UcSoqRiw3JCQiM3ErKysrRGQjKioqRi8kIjNPb3NDIyoqWypcKipGLDckJCIzLysrK1NUKD5BIkYsJCIzJXlSJSlmOWVdIyoqRiw3JCQiMyYpKioqKioqSGJhVzlGLCQiM2kzOnNnVjYmKikqRiw3JCQiMzsrKytJIilIbTtGLCQiM1lOXSIpZV8+ZykqRiw3JCQiMysrKytdSEx6PUYsJCIzJFsjPiUzJXoiPSMpKkYsNyQkIjMoKioqKioqKkgjSCc0QEYsJCIzLWloIilSMSVceCpGLDckJCIzKCkqKioqKioqRz0iSEJGLCQiM3IiZk0qZiZ5XHMqRiw3JCQiMzMrKyshPT8sYyNGLCQiMz1AcTYpKWZ0bScqRiw3JCQiMzMrKys/REJwRkYsJCIzJVEyRCkqUj8qMycqRiw3JCQiMzMrKys1M00rSUYsJCIzOEBGcEZbR1ImKkYsNyQkIjN3KioqKioqcEFxQEtGLCQiM1tXdiF5YTxvWSpGLDckJCIzPysrK1NcX1VNRiwkIjMhb2QmM2g1eClRKkYsNyQkIjMpKSoqKioqKnBdPHRPRiwkIjMiKUdjYUdoJjRJKkYsNyQkIjM1KysrXUFoJilRRiwkIjNcIipweXUhSFVAKkYsNyQkIjMhKSoqKioqKiopcDwuVEYsJCIzNiYqKnonUmtVPiIqRiw3JCQiM0QrKys/VlJWVkYsJCIzQWFTdmRGXDIhKkYsNyQkIjNJKysrSTkqM2MlRiwkIjNXJkchNF40TSoqKSlGLDckJCIzPSsrK3FISyR5JUYsJCIzLyF6aiMqPSp5InkpRiw3JCQiMy8rKys1LGA0XUYsJCIzYW9fRUlbdWEnKUYsNyQkIjMlKSoqKioqKipbZXVAJkYsJCIzeC1vbEIoNDVgKUYsNyQkIjM+KysrNU01UmFGLCQiM1s7Qlx2TFYiUilGLDckJCIzZSoqKioqKmY2MWZjRiwkIjMpPiU0dikpb3BXIylGLDckJCIzLSsrKyFlWlYqZUYsJCIzU19Kc18pZiJ5ISlGLDckJCIzLisrK113KD41J0YsJCIzXkIxdUE4XEF6Riw3JCQiMycpKioqKioqXCdSOk0nRiwkIjNOWmVWYnoxS3hGLDckJCIzWSsrKyEpZXNkbEYsJCIzRyJ6XDwlW2VcdkYsNyQkIjNxKioqKioqZjNqcm5GLCQiM2VtJD52bUgkZXRGLDckJCIzeSoqKioqKkh1ayxxRiwkIjMzW3ZaXksiKVJyRiw3JCQiMysrKytxV2ZLc0YsJCIzQjE+Y2VTdzBwRiw3JCQiM1krKyshM0FGVyhGLCQiM2NQYltnUnd5bUYsNyQkIjNXKioqKioqZiNvZW0oRiwkIjNhJHp6YW8xOVUnRiw3JCQiM20qKioqKipIcmokKXlGLCQiM0kzLCF5UyU0X2hGLDckJCIzTSsrKyEpM3E+IilGLCQiM1JUWDJmLypwJGVGLDckJCIzQCsrK107J1JLKUYsJCIzSmRvYyVSNT1hJkYsNyQkIjMhKSoqKioqKnpEY2UmKUYsJCIzNyJIOlBfUEA8JkYsNyQkIjNNKysrIXkqNHooKUYsJCIzL3QnPlQ/ZCMpeSVGLDckJCIzQSsrK1NvXSgqKilGLCQiM2EzI3prI0gva1ZGLDckJCIzRCsrK1N3MzsjKkYsJCIzWU0qR1gmPj8iKVFGLDckJCIzcCoqKioqKj5pLVIlKkYsJCIzTyhwQElDUkFJJEYsNyQkIjNzKioqKioqKnoiPWQmKkYsJCIzNVk1JVxwTUclSEYsNyQkIjN3KioqKioqenRMdicqRiwkIjMhXGUnemMqPXVfI0YsNyQkIjNXKysrdlduJHkqRiwkIjNtUWhbMnB1bz9GLDckJCIzKSoqKioqKipwOiwjKikqRiwkIjMlZnVRUUBVY1kiRiw3JCQiMyopKioqKipwQVooPSoqRiwkIjM9Y0lpPFg9czdGLDckJCIzcyoqKioqSClHW1gqKkYsJCIzVjBHLWMveFU1Riw3JCQiM2sqKioqKipSJj1BKCoqRiwkIjNDd2cmKj1dKClbdUYvNyQkIjNbKysrJj5hKikqKipGLCQiMydSQzQpUkA/WTlGLzckJSVGQUlMR0ZlamwtJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkIiIhRlxbbUZdW20tJStBWEVTTEFCRUxTRzYkUSJ4NiJRIUZjW20tJSpMSU5FU1RZTEVHNiNGXlttLUZnamw2IyUlTk9ORUctJSVWSUVXRzYkOyQhIzZGXFttJCIjNkZcW207JCExS1giWyNwT1pfRiwkIjJ5TW1lIzNrPjUhIzstJSxPUklFTlRBVElPTkc2JCQiI1hGXlttRlxdbS0lJUZPTlRHNiQlKkhFTFZFVElDQUdGW1ttLSUoU0NBTElOR0c2IyUsQ09OU1RSQUlORURHLSUrUFJPSkVDVElPTkc2I0ZqamwtJSpHUklEU1RZTEVHNiMlLFJFQ1RBTkdVTEFSRw== Maple's dsolve can solve this differential equation. It must be given to dsolve in the form NiMvLSUlZGlmZkc2JC0lInlHNiMlInhHRiosJComRioiIiJGJyEiIkYu . de := diff(y(x),x)=-x/y(x);
dsolve(de,y(x)); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiJGKSEiIkYw NiQvLSUieUc2IyUieEcqJC0lJXNxcnRHNiMsJiokKUYnIiIjIiIiISIiJSRfQzFHRjBGMC9GJCwkRihGMQ== Note that arbitrary constants appear in the form NiUlJF9DMUclJF9DMkclJF9DM0c= etc. To solve the differential equation with the initial condition NiMvLSUieUc2IyIiISIiIg== use dsolve , with a set of two equations as the first argument as follows. de := diff(y(x),x)=-x/y(x);
ic := y(0)=1;
soln := dsolve({de,ic},y(x)); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiJGKSEiIkYw NiM+JSNpY0cvLSUieUc2IyIiISIiIg== NiM+JSVzb2xuRy8tJSJ5RzYjJSJ4RyokLSUlc3FydEc2IywmKiQpRikiIiMiIiIhIiJGMkYyRjI=
We can graph this solution by setting up a function using the Maple routine rhs , which selects the expression on the right hand side of the equation, and the procedure unapply which converts the expression into a function. g := unapply(rhs(soln),x);
plot(g(x),x=-1.1..1.1,labels=[`x`,`y(x)`],scaling=constrained); NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCokLSUlc3FydEc2IywmKiQpOSQiIiMiIiIhIiJGNUY1RjVGKEYoRig= 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 ; The equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEdGJiUieUdGKEYo has the general form NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUiZkc2JCUieEclInlH .The function NiMvLSUiZkc2JCUieEclInlHLCQqJkYnIiIiRighIiJGLA== can be thought of as giving the gradient of any solution curve at the point NiMtJSFHNiQlInhHJSJ5Rw== . For example, at the pointNiMtJSFHNiQiIiMiIiI= , where NiMvJSJ4RyIiIw== and NiMvJSJ5RyIiIg== , the gradient of a solution curve passing through this point is NiMvLCQqJiUieEciIiIlInlHISIiRiksJComIiIjRidGJ0YpRik= NiMvJSFHLCQiIiMhIiI= , and we could draw a short line passing through the pointNiMtJSFHNiQiIiMiIiI= having a gradient of -2. Doing this for all points with integer coordinates (except those with y coordinate 0) between -3 and 3 gives the following picture. 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 The solution curve passing through the pointNiMtJSFHNiQiIiMiIiI= is NiMvLCYqJCUieEciIiMiIiIqJCUieUdGJ0YoIiIm , and this curve can be added to the picture. 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 In the following picture the gradient field is shown by using arrows instead of lines. It is drawn by using the Maple procedure DEplot in the package DEtools . DEplot draws arrows to indicate the direction in which a solution curve passing through a given point will procede (for increasing x ).de := diff(y(x),x)=-x/y(x);
DEtools[DEplot](de,y(x),x=-2..2,y=-2..2,arrows=medium,
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The solution NiMvJSJ5Ry0lJXNxcnRHNiMsJiIiIkYpKiQlInhHIiIjISIi can be plotted together with the gradient field. Even though it is unlikely that the curve NiMvJSJ5Ry0lJXNxcnRHNiMsJiIiIkYpKiQlInhHIiIjISIi , will pass exactly through more than the odd one or two points at which the arrows have been drawn, the curve follows the general direction of "flow" suggested by the arrows. de := diff(y(x),x)=-x/y(x);
ic := y(0)=1;
soln := dsolve({de,ic},y(x));
g := unapply(rhs(soln),x):
p1 := plot(g(x),x=-1..1,thickness=2,numpoints=75):
p2 := DEtools[DEplot](de,y(x),x=-1.2..1.2,y=0..1.2,arrows=medium,
color=COLOR(RGB,.5,0,1),scaling=constrained):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiJGKSEiIkYw NiM+JSNpY0cvLSUieUc2IyIiISIiIg== NiM+JSVzb2xuRy8tJSJ5RzYjJSJ4RyokLCYqJClGKSIiIyIiIiEiIkYvRi8jRi9GLg== 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; <Text-field layout="Heading 2" style="Heading 2">Examples</Text-field> ; <Text-field layout="Heading 2" style="Heading 2">Example 1</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiKigiIiNGJiUieEdGJiUieUdGJg== . (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiISIiIw== . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkJSJ5RyEiIg== NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYiIiNGJiUieEdGJg== . Thus NiMvLSUkSW50RzYkKiYiIiJGKCUieUchIiJGKS1GJTYkKiYiIiNGKCUieEdGKEYv . This gives NiMvLSUjbG5HNiMtJSRhYnNHNiMlInlHLCYqJCUieEciIiMiIiIlImNHRi8= . Then NiMvLSUkYWJzRzYjJSJ5Ry0lJGV4cEc2IywmKiQlInhHIiIjIiIiJSJjR0Yv , so that y = + NiMqJi0lJGV4cEc2IyUiY0ciIiItRiU2IyokJSJ4RyIiI0Yo . We can write this general solution in the form: NiMvJSJ5RyomJSJBRyIiIi0lJGV4cEc2IyokJSJ4RyIiI0Yn ------- (i), where A = + NiMtJSRleHBHNiMlImNH . de := diff(y(x),x)=2*x*y(x);
dsolve(de); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJCooIiIjIiIiRixGMEYpRjBGMA== NiMvLSUieUc2IyUieEcqJiUkX0MxRyIiIi0lJGV4cEc2IyokKUYnIiIjRipGKg== ; (b) Given the initial condition NiMvLSUieUc2IyIiISIiIw== , we substitute NiMvJSJ4RyIiIQ== and NiMvJSJ5RyIiIw== in the equation NiMvJSJ5RyomJSJBRyIiIi0lJGV4cEc2IyokJSJ4RyIiI0Yn to find the constant A . This gives NiMvIiIjKiYlIkFHIiIiLSUkZXhwRzYjIiIhRic= , so that NiMvJSJBRyIiIw== . The corresponding solution is: NiMvJSJ5RyomIiIjIiIiLSUkZXhwRzYjKiQlInhHRiZGJw== . It is easy check that this solution is correct by differentiation. Given NiMvJSJ5RyomIiIjIiIiLSUkZXhwRzYjKiQlInhHRiZGJw== , NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYiIiNGJi0lJGV4cEc2IyokJSJ4R0YqRiY= NiMqJiUiZEciIiIlI2R4RyEiIg== NiM3IyokJSJ4RyIiIw== NiMvJSFHKioiIiMiIiItJSRleHBHNiMqJCUieEdGJkYnRiZGJ0YsRic= NiMvJSFHKigiIiUiIiIlInhHRictJSRleHBHNiMqJEYoIiIjRic= . On the other hand, substituting NiMvJSJ5RyomIiIjIiIiLSUkZXhwRzYjKiQlInhHRiZGJw== in the right side of the differential equation gives: NiMvKigiIiMiIiIlInhHRiYlInlHRiYqKkYlRiYlIi5HRiZGJUYmLSUkZXhwRzYjKiRGJ0YlRiY= NiMvJSFHKiYiIiUiIiItJSRleHBHNiMqJCUieEciIiNGJw== , also. de := diff(y(x),x)=2*x*y(x);
ic := y(0)=2;
dsolve({de,ic},y(x));
g := unapply(rhs(%),x):
p1 := plot(g(x),x=-1.1..1.1,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=-1.1..1.1,y=0..6.35,
arrows=medium,color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJCooIiIjIiIiRixGMEYpRjBGMA== NiM+JSNpY0cvLSUieUc2IyIiISIiIw== NiMvLSUieUc2IyUieEcsJComIiIjIiIiLSUkZXhwRzYjKiQpRidGKkYrRitGKw== 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; <Text-field layout="Heading 2" style="Heading 2">Example 2</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiKigiIiRGJiokJSJ4RyIiI0YmKiZGLUYmJSJ5R0YmRig= . (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiIiIiJA== . Solution : (a) The variables can be separated giving: NiMqJiIiIyIiIiUieUdGJQ== NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYiIiRGJiokJSJ4RyIiI0Ym . Thus NiMvLSUkSW50RzYkKiYiIiMiIiIlInlHRilGKi1GJTYkKiYiIiRGKSokJSJ4R0YoRilGMA== . This gives NiMvKiQlInlHIiIjLCYqJCUieEciIiQiIiIlImNHRis= . The general solution of the differential equation is: NiMvKiQlInlHIiIjLCYqJCUieEciIiQiIiIlImNHRis= in implicit form. Solving for y gives the explicit general solutions: NiMvJSJ5Ry0lJXNxcnRHNiMsJiokJSJ4RyIiJCIiIiUiY0dGLA== , NiMvJSJ5RywkLSUlc3FydEc2IywmKiQlInhHIiIkIiIiJSJjR0YtISIi . de := diff(y(x),x)=3*x^2/(2*y(x));
dsolve(de); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiNGKSEiIiMiIiRGLw== NiQvLSUieUc2IyUieEcqJC0lJXNxcnRHNiMsJiokKUYnIiIkIiIiRjAlJF9DMUdGMEYwL0YkLCRGKCEiIg== ; (b) Given the initial condition NiMvLSUieUc2IyIiIiIiJA== , we substitute NiMvJSJ4RyIiIg== and NiMvJSJ5RyIiJA== in the equation NiMvKiQlInlHIiIjLCYqJCUieEciIiQiIiIlImNHRis= to find the constant c . This gives NiMvIiIqLCYiIiJGJiUiY0dGJg== , so that NiMvJSJjRyIiKQ== . The corresponding solution is: NiMvJSJ5Ry0lJXNxcnRHNiMsJiokJSJ4RyIiJCIiIiIiKUYs . Note that the equation NiMvJSJ5RywkLSUlc3FydEc2IywmKiQlInhHIiIkIiIiIiIpRi0hIiI= gives NiMvJSJ5RywkIiIkISIi when NiMvJSJ4RyIiIg== , and so the initial condition is not satisfied. de := diff(y(x),x)=3*x^2/(2*y(x));
ic := y(1)=3;
dsolve({de,ic},y(x));
g := unapply(rhs(%),x):
p1 := plot(g(x),x=-2..4,thickness=2,numpoints=75):
p2 := DEtools[DEplot](de,y(x),x=-2.5..4,y=0..8,
arrows=medium,color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComIyIiJCIiIyIiIiomRixGMUYpISIiRjJGMg== NiM+JSNpY0cvLSUieUc2IyIiIiIiJA== NiMvLSUieUc2IyUieEcqJCwmKiQpRiciIiQiIiJGLSIiKUYtI0YtIiIj 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; <Text-field layout="Heading 2" style="Heading 2">Example 3</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMqJCUieEciIiQ= NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYiIiNGJiokJSJ5R0YqRiY= . (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiIiomIiIlRiciIiYhIiI= . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkKiQlInlHIiIjISIi NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYiIiNGJiokJSJ4RyIiJEYo . Thus NiMvLSUkSW50RzYkKSUieUcsJCIiIyEiIkYoLUYlNiQqJkYqIiIiKSUieEcsJCIiJEYrRi9GMQ== . This gives NiMvLCQqJiIiIkYmJSJ5RyEiIkYoLCYqJkYmRiYqJCUieEciIiNGKEYoJSJjR0Ym ------- (i), so NiMvLCQqJiIiIkYmJSJ5RyEiIkYoKiYsJkYmRigqJiUiY0dGJiokJSJ4RyIiI0YmRiZGJkYtRig= . The general solution of the differential equation is NiMvJSJ5RyomJSJ4RyIiIywmIiIiRikqJiUiY0dGKSokRiZGJ0YpISIiRi0= . de := x^3*diff(y(x),x)=2*y(x)^2;
dsolve(de); NiM+JSNkZUcvKiYpJSJ4RyIiJCIiIi0lJWRpZmZHNiQtJSJ5RzYjRihGKEYqLCQqJiIiI0YqKUYuRjNGKkYq NiMvLSUieUc2IyUieEcqJkYnIiIjLCYiIiJGKyomJSRfQzFHRispRidGKUYrRishIiI= ; (b) Given the initial condition NiMvLSUieUc2IyIiIiomIiIlRiciIiYhIiI= , we substitute NiMvJSJ4RyIiIg== and NiMvJSJ5RyomIiIlIiIiIiImISIi in equation (i) to find the constant c . This gives NiMvLCQqJiIiJiIiIiIiJSEiIkYpLCZGJ0YpJSJjR0Yn , so that NiMvJSJjRywkKiYiIiJGJyIiJSEiIkYp . The corresponding solution is: NiMvJSJ5RyomJSJ4RyIiIywmIiIiRikqJkYmRiciIiUhIiJGKUYs or NiMvJSJ5RyooIiIlIiIiKiQlInhHIiIjRicsJkYmRidGKEYnISIi . We can check this solution as follows. NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYsJioqIiIpRiYlInhHRiYlIi5HRiYsJiIiJUYmKiRGLSIiI0YmRiZGJiosRjBGJkYxRiZGLkYmRjJGJkYtRiZGKEYmKiRGL0YyRig= NiMvJSFHKigiI0siIiIlInhHRicqJCwmIiIlRicqJEYoIiIjRidGLSEiIg== , so that substituting for NiMqJiUjZHlHIiIiJSNkeEchIiI= in left side of the differential equation gives: NiMqJCUieEciIiQ= NiMvKiYlI2R5RyIiIiUjZHhHISIiKigiI0tGJiokJSJ4RyIiJUYmKiQsJkYtRiYqJEYsIiIjRiZGMUYo . On the other hand, substituting for y from the solution in the expression on the right side of the differential equation gives: NiMvKiYiIiMiIiIqJCUieUdGJUYmKiZGJUYmKiQqKCIiJUYmKiQlInhHRiVGJiwmRixGJkYtRiYhIiJGJUYm NiMvJSFHKigiI0siIiIqJCUieEciIiVGJyokLCZGKkYnKiRGKSIiI0YnRi4hIiI= also. Since the particular solution NiMvJSJ5RyooIiIlIiIiKiQlInhHIiIjRicsJkYmRidGKEYnISIi also clearly satisfies the initial condition, the last calculation verifies that this solution is correct. As a technical point, the working used to obtain the solution is not valid when NiMvJSJ4RyIiIQ== and/or NiMvJSJ5RyIiIQ== , because these values lead to division by zero. However, our checking shows that the solution is valid when NiMvJSJ4RyIiIQ== (and NiMvJSJ5RyIiIQ== ). de := x^3*diff(y(x),x)=2*y(x)^2;
ic := y(1)=4/5;
dsolve({de,ic},y(x)):
normal(%);
g := unapply(rhs(%),x):
p1 := plot(g(x),x=-6..6,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=-6..6,y=0..4,
arrows=medium,color=COLOR(RGB,.5,0,1)):
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; <Text-field layout="Heading 2" style="Heading 2">Example 4</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiKiglInlHRiYlJGNvc0dGJiUieEdGJg== (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiISIiIg== . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkJSJ5RyEiIg== NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlJGNvc0dGJiUieEdGJg== . Thus NiMvLSUkSW50RzYkKiYiIiJGKCUieUchIiJGKS1GJTYkKiYlJGNvc0dGKCUieEdGKEYv . This gives NiMvLSUjbG5HNiMtJSRhYnNHNiMlInlHLCYqJiUkc2luRyIiIiUieEdGLkYuJSJjR0Yu , so y = + NiMqJi0lJGV4cEc2IyUiY0ciIiItRiU2IyomJSRzaW5HRiglInhHRihGKA== , or NiMvJSJ5RyomJSJBRyIiIi0lJGV4cEc2Iy0lJHNpbkc2IyUieEdGJw== , where NiMlIkFH = + NiMtJSRleHBHNiMlImNH . de := diff(y(x),x)=y(x)*cos(x);
dsolve(de); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJkYpIiIiLSUkY29zR0YrRi4= NiMvLSUieUc2IyUieEcqJiUkX0MxRyIiIi0lJGV4cEc2Iy0lJHNpbkdGJkYq ; (b) Given the initial condition NiMvLSUieUc2IyIiISIiIg== , we substitute NiMvJSJ4RyIiIQ== and NiMvJSJ5RyIiIg== in the equation NiMvJSJ5RyomJSJBRyIiIi0lJGV4cEc2Iy0lJHNpbkc2IyUieEdGJw== to find the constant A . This gives NiMvIiIiKiYlIkFHRiQtJSRleHBHNiMiIiFGJA== , so that NiMvJSJBRyIiIg== . The corresponding solution is: NiMvJSJ5Ry0lJGV4cEc2Iy0lJHNpbkc2IyUieEc= . de := diff(y(x),x)=y(x)*cos(x);
ic := y(0)=1;
dsolve({de,ic},y(x));
g := unapply(rhs(%),x):
p1 := plot(g(x),x=-5..10,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=-5..10,y=0..3,arrows=medium,
dirgrid=[30,25],color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJkYpIiIiLSUkY29zR0YrRi4= NiM+JSNpY0cvLSUieUc2IyIiISIiIg== NiMvLSUieUc2IyUieEctJSRleHBHNiMtJSRzaW5HRiY= 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; <Text-field layout="Heading 2" style="Heading 2">Example 5</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUlc3FydEc2IyomJSJ4R0YmJSJ5R0Ym (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiISIiIg== . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkLSUlc3FydEc2IyUieUchIiI= NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUlc3FydEc2IyUieEc= . Thus NiMvLSUkSW50RzYkKiYiIiJGKC0lJXNxcnRHNiMlInlHISIiRiwtRiU2JC1GKjYjJSJ4R0Yy , or NiMvLSUkSW50RzYkKSUieUcsJComIiIiRisiIiMhIiJGLUYoLUYlNiQpJSJ4R0YqRjE= . This gives NiMvKiYiIiMiIiIpJSJ5RyomRiZGJkYlISIiRiYqJkYlRiYiIiRGKg== NiMsJiklInhHKiYiIiQiIiIiIiMhIiJGKCUiY0dGKA== . or NiMvLSUlc3FydEc2IyUieUcqJiIiIkYpIiIkISIi NiMsJiklInhHKiYiIiQiIiIiIiMhIiJGKCYlImNHNiNGKEYo , where NiMvJiUiY0c2IyIiIiomRiVGJyIiIyEiIg== . Then on squaring both sides of the last equation we obtain: NiMvJSJ5RywmKiYlInhHIiIkIiIqISIiIiIiKigiIiNGKyYlImNHNiNGK0YrRihGKkYr NiMsJiklInhHKiYiIiQiIiIiIiMhIiJGKCokJiUiY0c2I0YoRilGKA== . de := diff(y(x),x)=sqrt(x)*sqrt(y(x));
dsolve(de);
solve(%,y(x)); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJkYsIyIiIiIiI0YpRi4= NiMvLCgqJC0lInlHNiMlInhHIyIiIiIiI0YrKiYiIiQhIiJGKSNGLkYsRi8lJF9DMUdGLyIiIQ== NiMsKComIiIqISIiJSJ4RyIiJCIiIioqIiIjRilGKEYmRicjRihGKyUkX0MxR0YpRikqJClGLUYrRilGKQ== ; (b) Given the initial condition NiMvLSUieUc2IyIiISIiIg== , we substitute NiMvJSJ4RyIiIQ== and NiMvJSJ5RyIiIg== in equation (i) to give NiMvJiUiY0c2IyIiIkYn . The corresponding solution is: NiMvJSJ5RywmKiYlInhHIiIkIiIqISIiIiIiKiYiIiNGK0YoRipGKw== NiMsJiklInhHKiYiIiQiIiIiIiMhIiJGKEYoRig= . de := diff(y(x),x)=sqrt(x)*sqrt(y(x));
ic := y(0)=1;
dsolve({de,ic},y(x)):
allvalues(%);
g := unapply(rhs(%),x):
p1 := plot(g(x),x=0..3,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=0..3,y=0..7.5,
arrows=medium,color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJkYsIyIiIiIiI0YpRi4= NiM+JSNpY0cvLSUieUc2IyIiISIiIg== NiMvLSUieUc2IyUieEcsKComIiIqISIiRiciIiQiIiIqKCIiI0YtRixGK0YnI0YsRi9GLUYtRi0= 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; <Text-field layout="Heading 2" style="Heading 2">Example 6</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMqJiIiIyIiIi0lJXNxcnRHNiMlInhHRiU= NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlInlHRiYsJkYqRiZGJkYoRiY= (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiISomIiIiRikiIiMhIiI= . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkKiYlInlHRiQsJkYmRiRGJCEiIkYkRig= NiMvKiYlI2R5RyIiIiUjZHhHISIiKiZGJkYmKiYiIiNGJi0lJXNxcnRHNiMlInhHRiZGKA== . Hence NiMvLSUkSW50RzYkKiYiIiJGKComJSJ5R0YoLCZGKkYoRighIiJGKEYsRiotRiU2JComRihGKComIiIjRigtJSVzcXJ0RzYjJSJ4R0YoRixGNQ== ------- (i). The expression NiMqJiIiIkYkKiYlInlHRiQsJkYmRiRGJCEiIkYkRig= has a partial fraction expansion of the form: NiMvKiYiIiJGJSomJSJ5R0YlLCZGJ0YlRiUhIiJGJUYpLCYqJiUiQUdGJUYnRilGJSomJSJCR0YlRihGKUYl . It is sufficient to find constants A and B so that: NiMvIiIiLCYqJiUiQUdGJCwmJSJ5R0YkRiQhIiJGJEYkKiYlIkJHRiRGKUYkRiQ= . Letting NiMvJSJ5RyIiIQ== gives NiMvIiIiKiglIkFHRiQlIi5HRiQsJEYkISIiRiQ= , so that NiMvJSJBRywkIiIiISIi . Letting NiMvJSJ5RyIiIg== shows that NiMvJSJCRyIiIg== . Hence NiMvKiYiIiJGJSomJSJ5R0YlLCZGJ0YlRiUhIiJGJUYpLCYqJkYlRiVGKEYpRiUqJkYlRiVGJ0YpRik= so that (i) becomes: NiMvLSUkSW50RzYkLSUhRzYjLCYqJiIiIkYsLCYlInlHRixGLCEiIkYvRiwqJkYsRixGLkYvRi9GLiomRixGLCIiI0Yv NiMtJSRJbnRHNiQpJSJ4RywkKiYiIiJGKiIiIyEiIkYsRic= . This gives NiMvLCYtJSNsbkc2Iy0lJGFic0c2IywmJSJ5RyIiIkYtISIiRi0tRiY2Iy1GKTYjRixGLiwmKSUieEcqJkYtRi0iIiNGLkYtJSJjR0Yt , or NiMvLSUjbG5HNiMtJSRhYnNHNiMqJiwmJSJ5RyIiIkYtISIiRi1GLEYuLCYtJSVzcXJ0RzYjJSJ4R0YtJSJjR0Yt . Then NiMvLSUkYWJzRzYjKiYsJiUieUciIiJGKiEiIkYqRilGKyomLSUkZXhwRzYjLSUlc3FydEc2IyUieEdGKi1GLjYjJSJjR0Yq , or NiMvKiYsJiUieUciIiJGJyEiIkYnRiZGKComJSJBR0YnLSUkZXhwRzYjLSUlc3FydEc2IyUieEdGJw== , where NiMvJSJBRyUhRw== + NiMtJSRleHBHNiMlImNH , or NiMvLCYiIiJGJSomRiVGJSUieUchIiJGKComJSJBR0YlLSUkZXhwRzYjLSUlc3FydEc2IyUieEdGJQ== ------- (ii). Then NiMvKiYiIiJGJSUieUchIiIsJkYlRiUqJiUiQUdGJS0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHRiVGJw== so that NiMvJSJ5RyomIiIiRiYsJkYmRiYqJiUiQUdGJi0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHRiYhIiJGMQ== . de := 2*sqrt(x)*diff(y(x),x)=y(x)*(y(x)-1);
dsolve(de); NiM+JSNkZUcvLCQqKCIiIyIiIiUieEcjRilGKC0lJWRpZmZHNiQtJSJ5RzYjRipGKkYpRikqJkYvRiksJkYvRilGKSEiIkYp NiMvLSUieUc2IyUieEcqJiIiIkYpLCZGKUYpKiYtJSRleHBHNiMqJEYnI0YpIiIjRiklJF9DMUdGKUYpISIi ; (b) Given the initial condition NiMvLSUieUc2IyIiISomIiIiRikiIiMhIiI= , we substitute NiMvJSJ4RyIiIQ== and NiMvJSJ5RyomIiIiRiYiIiMhIiI= in equation (ii) to find the constant A . This gives NiMvLCYiIiJGJSIiIyEiIiomJSJBR0YlLSUkZXhwRzYjIiIhRiU= , so that NiMvJSJBRywkIiIiISIi . The corresponding solution is: NiMvJSJ5RyomIiIiRiYsJkYmRiYtJSRleHBHNiMtJSVzcXJ0RzYjJSJ4R0YmISIi . We can check this solution as follows. Writing NiMvJSJ5RyksJiIiIkYnLSUkZXhwRzYjLSUlc3FydEc2IyUieEdGJywkRichIiI= we see that NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYsJEYmRihGJiksJkYmRiYtJSRleHBHNiMtJSVzcXJ0RzYjJSJ4R0YmLCQiIiNGKEYm NiMqJiUiZEciIiIlI2R4RyEiIg== NiM3IywmIiIiRiUtJSRleHBHNiMtJSVzcXJ0RzYjJSJ4R0Yl NiMvJSFHLCQqJiksJiIiIkYpLSUkZXhwRzYjLSUlc3FydEc2IyUieEdGKSwkIiIjISIiRilGKkYpRjM= NiMqJiUiZEciIiIlI2R4RyEiIg== NiM3Iy0lJXNxcnRHNiMlInhH NiMvJSFHLCQqJiksJiIiIkYpLSUkZXhwRzYjLSUlc3FydEc2IyUieEdGKSwkIiIjISIiRilGKkYpRjM= NiMqJiIiIkYkKiYiIiNGJC0lJXNxcnRHNiMlInhHRiQhIiI= NiMvJSFHLCQqJi0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHIiIiKigiIiNGLkYqRi4sJkYuRi5GJ0YuRjAhIiJGMg== . Substituting for NiMqJiUjZHlHIiIiJSNkeEchIiI= in left side of the differential equation gives: NiMqJiIiIyIiIi0lJXNxcnRHNiMlInhHRiU= NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJi0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHRiYqJCwmRiZGJkYrRiYiIiNGKEYo . On the other hand, substituting for y from the solution in the expression on the right side of the differential equation gives: NiMvKiYlInlHIiIiLCZGJUYmRiYhIiJGJiomRiZGJiwmRiZGJi0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHRiZGKA== NiMtJSFHNiMsJiomIiIiRigsJkYoRigtJSRleHBHNiMtJSVzcXJ0RzYjJSJ4R0YoISIiRihGKEYx NiMvJSFHKiYiIiJGJiwmRiZGJi0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHRiYhIiI= NiMtJSFHNiMsJComLSUkZXhwRzYjLSUlc3FydEc2IyUieEciIiIsJkYvRi9GKEYvISIiRjE= NiMvJSFHLCQqJi0lJGV4cEc2Iy0lJXNxcnRHNiMlInhHIiIiKiQsJkYuRi5GJ0YuIiIjISIiRjI= also. Since the particular solution NiMvJSJ5RyomIiIiRiYsJkYmRiYtJSRleHBHNiMtJSVzcXJ0RzYjJSJ4R0YmISIi also clearly satisfies the initial condition, the last calculation verifies that this solution is correct. de := 2*sqrt(x)*diff(y(x),x)=y(x)*(y(x)-1);
ic := y(0)=1/2;
dsolve({de,ic},y(x));
g := unapply(rhs(%),x):
p1 := plot(g(x),x=0..8,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=0..8,y=0..0.5,
arrows=medium,color=COLOR(RGB,.5,0,1)):
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"**R"3'>]!#:Fd]lFa]l-%+AXESLABELSG6'Q"x6"Q%y(x)Fgbjl-%%FONTG6$%*HELVETICAGFb]l%+HORIZONTALGF]cjl-%*LINESTYLEG6#F+-F^]l6#%%NONEG-%%VIEWG6$;$!"%Fc]l$"#%)Fc]l;$!#D!"$$"$D&F_djl-%,ORIENTATIONG6$$"#XF+FedjlFibjl-%+PROJECTIONG6#Fa]l-%*GRIDSTYLEG6#%,RECTANGULARG ; <Text-field layout="Heading 2" style="Heading 2">Example 7</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMlInhH NiMvKiYlI2R5RyIiIiUjZHhHISIiLCZGJkYmKiQlInlHIiIjRiY= (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiIkYn . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkLCZGJEYkKiQlInlHIiIjRiQhIiI= NiMvKiYlI2R5RyIiIiUjZHhHISIiKiZGJkYmJSJ4R0Yo . Thus NiMvLSUkSW50RzYkKiYiIiJGKCwmRihGKCokJSJ5RyIiI0YoISIiRistRiU2JComRihGKCUieEdGLUYx . This gives NiMvKiYlJ2FyY3RhbkciIiIlInlHRiYsJi0lI2xuRzYjLSUkYWJzRzYjJSJ4R0YmJSJjR0Ym ------- (i), or NiMvJSJ5Ry0lJHRhbkc2IywmLSUjbG5HNiMtJSRhYnNHNiMlInhHIiIiJSJjR0Yw . de := x*diff(y(x),x)=1+y(x)^2;
dsolve(de); NiM+JSNkZUcvKiYlInhHIiIiLSUlZGlmZkc2JC0lInlHNiNGJ0YnRigsJkYoRigqJClGLCIiI0YoRig= NiMvLSUieUc2IyUieEctJSR0YW5HNiMsJi0lI2xuR0YmIiIiJSRfQzFHRi4= ; (b) Given the initial condition NiMvLSUieUc2IyIiIkYn , we substitute NiMvJSJ4RyIiIg== and NiMvJSJ5RyIiIg== in equation (i) to find the constant A . This gives NiMvLSUnYXJjdGFuRzYjIiIiLCYtJSNsbkdGJkYnJSJjR0Yn . Since NiMvLSUnYXRjdGFuRzYjIiIiKiYlI1BpR0YnIiIlISIi and NiMvLSUjbG5HNiMiIiIiIiE= , we see that NiMvJSJjRyomJSNQaUciIiIiIiUhIiI= . The corresponding solution is: NiMvJSJ5Ry0lJHRhbkc2IywmLSUjbG5HNiMlInhHIiIiKiYlI1BpR0YtIiIlISIiRi0= . The absolute value operation has been quietly dropped to give the single branch of the solution curve shown in the following picture. de := x*diff(y(x),x)=1+y(x)^2;
ic := y(1)=1;
dsolve({de,ic},y(x));
g := unapply(rhs(%),x):
p1 := plot(g(x),x=0.095..2.193,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=0.095..2.193,y=-12..12,
arrows=medium,color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvKiYlInhHIiIiLSUlZGlmZkc2JC0lInlHNiNGJ0YnRigsJkYoRigqJClGLCIiI0YoRig= NiM+JSNpY0cvLSUieUc2IyIiIkYp NiMvLSUieUc2IyUieEctJSR0YW5HNiMsJi0lI2xuR0YmIiIiKiYiIiUhIiIlI1BpR0YuRi4= 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; <Text-field layout="Heading 2" style="Heading 2">Example 8</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMvKiYlI2R5RyIiIiUjZHhHISIiKiglInhHRiYlInlHRiYsJkYmRiYqJEYqIiIjRiZGKA== (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiISIiJA== . Solution : (a) The variables can be separated giving: NiMqJiIiIkYkJSJ5RyEiIg== NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlInhHRiYsJkYmRiYqJEYqIiIjRiZGKA== . Thus NiMvLSUkSW50RzYkKiYiIiJGKCUieUchIiJGKS1GJTYkKiYlInhHRigsJkYoRigqJEYuIiIjRihGKkYu . The integral NiMtJSRJbnRHNiQqJiUieEciIiIsJkYoRigqJEYnIiIjRighIiJGJw== can be found as follows by using the substitution NiMvJSJ1RywmIiIiRiYqJCUieEciIiNGJg== . NiMtJSRJbnRHNiQqJiUieEciIiIsJkYoRigqJEYnIiIjRighIiJGJw== ... NiMtJSpQSUVDRVdJU0VHNiU3JC8lInVHLCYiIiJGKiokJSJ4RyIiI0YqJSFHNyQvJSNkdUcqKkYtRipGLEYqJSIuR0YqJSNkeEdGKkYuNyQvKigtRi42IyomRipGKkYtISIiRipGM0YqRjFGKiooRixGKkYzRipGNEYqRi4= NiMvJSFHKiYiIiJGJiIiIyEiIg== NiMvLSUkSW50RzYkKiYiIiJGKCUidUchIiJGKSomRihGKCIiI0Yq NiMvLCYtJSNsbkc2IyUidUciIiIlImNHRikqJkYpRikiIiMhIiI= NiMsJi0lI2xuRzYjLCYiIiJGKCokJSJ4RyIiI0YoRiglImNHRig= . It follows that NiMvLSUjbG5HNiMtJSRhYnNHNiMlInlHKiYiIiJGLCIiIyEiIg== NiMsJi0lI2xuRzYjLCYqJCUieEciIiMiIiJGK0YrRislImNHRis= or NiMvLSUjbG5HNiMtJSRhYnNHNiMlInlHLCYtRiU2Iy0lJXNxcnRHNiMsJiIiIkYyKiQlInhHIiIjRjJGMiUiY0dGMg== . Hence NiMvLCYtJSNsbkc2Iy0lJGFic0c2IyUieUciIiItRiY2Iy0lJXNxcnRHNiMsJkYsRiwqJCUieEciIiNGLCEiIiUiY0c= or NiMvKiYtJSNsbkc2Iy0lJGFic0c2IyUieUciIiItJSVzcXJ0RzYjLCZGLEYsKiQlInhHIiIjRiwhIiIlImNH . This gives NiMvKiYtJSRhYnNHNiMlInlHIiIiLSUlc3FydEc2IywmRilGKSokJSJ4RyIiI0YpISIiLSUkZXhwRzYjJSJjRw== , that is, NiMvKiYlInlHIiIiLSUlc3FydEc2IywmRiZGJiokJSJ4RyIiI0YmISIiJSJBRw== , where NiMvJSJBRyUhRw== + NiMtJSRleHBHNiMlImNH . Thus we obtain the general solution: NiMvJSJ5RyomJSJBRyIiIi0lJXNxcnRHNiMsJkYnRicqJCUieEciIiNGJ0Yn . de := diff(y(x),x)=x*y(x)/(1+x^2);
dsolve(de); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqKEYsIiIiRilGLiwmRi5GLiokKUYsIiIjRi5GLiEiIg== NiMvLSUieUc2IyUieEcqJiUkX0MxRyIiIiwmRipGKiokKUYnIiIjRipGKiNGKkYu ; (b) Given the initial condition NiMvLSUieUc2IyIiISIiJA== , we substitute NiMvJSJ4RyIiIQ== and NiMvJSJ5RyIiJA== in the equation NiMvJSJ5RyomJSJBRyIiIi0lJXNxcnRHNiMsJkYnRicqJCUieEciIiNGJ0Yn to see that NiMvJSJBRyIiJA== . The corresponding solution is: NiMvJSJ5RyomIiIkIiIiLSUlc3FydEc2IywmRidGJyokJSJ4RyIiI0YnRic= . de := diff(y(x),x)=x*y(x)/(1+x^2);
ic := y(0)=3;
dsolve({de,ic},y(x));
g := unapply(rhs(%),x):
p1 := plot(g(x),x=-4..4,thickness=2):
p2 := DEtools[DEplot](de,y(x),x=-4..4,y=0..13,
arrows=medium,color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqKEYsIiIiRilGLiwmRi5GLiokKUYsIiIjRi5GLiEiIg== NiM+JSNpY0cvLSUieUc2IyIiISIiJA== NiMvLSUieUc2IyUieEcsJComIiIkIiIiLCZGK0YrKiQpRiciIiNGK0YrI0YrRi9GKw== 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; <Text-field layout="Heading 2" style="Heading 2">Example 9</Text-field> ; Question : (a) Find the general solution of the differential equation: NiMtJSFHNiMsJiomIiIkIiIiKiQlInlHIiIjRilGKUYsRik= NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlJHNpbkciIiMlInhHRiY= (b) Find the particular solution of the differential equation given in (a) subject to the initial condtion NiMvLSUieUc2IyIiIUYn . Solution : (a) The variables in the differential equation NiMtJSFHNiMsJiomIiIkIiIiKiQlInlHIiIjRilGKUYsRik= NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlJHNpbkciIiMlInhHRiY= are already separated. Thus NiMvLSUkSW50RzYkLSUhRzYjLCYqJiIiJCIiIiokJSJ5RyIiI0YtRi1GMEYtRi8tRiU2JComJSRzaW5HRjAlInhHRi1GNQ== , that is, NiMvLSUkSW50RzYkLSUhRzYjLCYqJiIiJCIiIiokJSJ5RyIiI0YtRi1GMEYtRi8tRiU2JCwmKiZGLUYtRjAhIiJGLSoqJSRjb3NHRi1GMEYtJSJ4R0YtRjBGNUY1Rjg= , It follows that NiMvLCYqJCUieUciIiQiIiIqJiIiI0YoRiZGKEYoLCgqJiUieEdGKEYqISIiRigqKiUkc2luR0YoRipGKEYtRigiIiVGLkYuJSJjR0Yo ------- (i). which is the general solution in implicit form. de := (3*y(x)^2+2)*diff(y(x),x)=sin(x)^2;
dsolve(de,implicit); NiM+JSNkZUcvKiYsJiomIiIkIiIiKS0lInlHNiMlInhHIiIjRipGKkYwRipGKi0lJWRpZmZHNiRGLEYvRioqJCktJSRzaW5HRi5GMEYq NiMvLCwqJiIiIyEiIiUieEciIiJGKSomI0YpIiIlRiktJSRzaW5HNiMsJComRiZGKUYoRilGKUYpRicqJCktJSJ5RzYjRigiIiRGKUYnKiZGJkYpRjRGKUYnJSRfQzFHRikiIiE= ; (b) Given the initial condition NiMvLSUieUc2IyIiIUYn , we substitute NiMvJSJ4RyIiIQ== and NiMvJSJ5RyIiIQ== in equation (i) to see that NiMvJSJjRyIiIQ== . The corresponding solution is: NiMvLCYqJCUieUciIiQiIiIqJiIiI0YoRiZGKEYoLCYqJiUieEdGKEYqISIiRigqKiUkc2luR0YoRipGKEYtRigiIiVGLkYu . p1 := plots[implicitplot](y^3+2*y=x/2-sin(2*x)/4,
x=-6..6,y=-1.4..1.4,grid=[30,20],thickness=2):
p2 := DEtools[DEplot](de,y(x),x=-6..6,y=-1.4..1.4,arrows=medium,
dirgrid=[25,20],color=COLOR(RGB,.5,0,1)):
plots[display]([p1,p2],labels=[`x`,`y(x)`]); 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; ; <Text-field layout="Heading 2" style="Heading 2">Getting Maple to show the steps of the solution</Text-field> ; We can perform the steps of the solution of a differential equation with separable variables as follows. Separate the variables and equate the integrals of the left and right sides, including an arbitrary constant. Use the unevaluated form Int first, and then try to compute the integrals. Try to solve for the dependent variable y to obtain explicit formulas for y in terms of x . Convert the solutions to equations of the form NiMvLSUieUc2IyUieEclJn4ufi5+Rw== . Example NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEdGJiUieUdGKEYo gives NiMvLSUkaW50RzYkJSJ5R0YnLCQtRiU2JCUieEdGKyEiIg== . C := 'C':
Int(y,y)=-Int(x,x)+C[1];
eq := value(%);
soln := [solve(eq,y)]:
op(map(_U -> y(x)=_U,soln)); NiMvLSUkSW50RzYkJSJ5R0YnLCYtRiU2JCUieEdGKyEiIiYlIkNHNiMiIiJGMA== NiM+JSNlcUcvLCQqJCklInlHIiIjIiIiI0YrRiosJiokKSUieEdGKkYrIyEiIkYqJiUiQ0c2I0YrRis= NiQvLSUieUc2IyUieEcqJC0lJXNxcnRHNiMsJiokKUYnIiIjIiIiISIiKiZGL0YwJiUiQ0c2I0YwRjBGMEYwL0YkLCRGKEYx ; We can obtain a particular solution curve passing through a specific point by performing additional steps. Substitute the coordinates in the equation which describes the general solution. Solve for the arbitrary constant. Substitute the value of the constant into the expressions for the solutions. eq2 := subs({x=0,y=2},eq);
const := solve(eq2,C[1]);
soln := subs(C[1]=const,soln);
op(map(_U -> y(x)=_U,soln)); NiM+JSRlcTJHLyIiIyYlIkNHNiMiIiI= NiM+JSZjb25zdEciIiM= NiM+JSVzb2xuRzckKiQtJSVzcXJ0RzYjLCYqJCklInhHIiIjIiIiISIiIiIlRi9GLywkRiZGMA== NiQvLSUieUc2IyUieEcqJC0lJXNxcnRHNiMsJiokKUYnIiIjIiIiISIiIiIlRjBGMC9GJCwkRihGMQ== If there is more than explicit solution for NiMtJSJ5RzYjJSJ4Rw== , check which solution fits the initial condition. Give the solution in the form NiMvLSUieUc2IyUieEclJn4ufi5+Rw== . y(x)=soln[1]; NiMvLSUieUc2IyUieEcqJC0lJXNxcnRHNiMsJiokKUYnIiIjIiIiISIiIiIlRjBGMA== A particular solution can also be found using dsolve . de := diff(y(x),x)=-x/y(x);
ic := y(0)=2;
dsolve({de,ic},y(x)); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiJGKSEiIkYw NiM+JSNpY0cvLSUieUc2IyIiISIiIw== NiMvLSUieUc2IyUieEcqJC0lJXNxcnRHNiMsJiokKUYnIiIjIiIiISIiIiIlRjBGMA== ; ; <Text-field layout="Normal257" style="Normal257">Code for pictures </Text-field> de := diff(y(x),x)=-x/y(x);
p1 := plot([[seq(seq([i,j],i=-3..3),j=-3..3)]$3],style=point,
symbol=[circle,diamond,cross],color=black):
p2 := DEtools[DEplot](de,y(x),x=-3..3,y=-3..3,
dirgrid=[7,7],arrows=line,color=COLOR(RGB,.5,0,1)):
t1 := plots[textplot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):
plots[display]([p1,p2,t1],scaling=constrained); de := diff(y(x),x)=-x/y(x);
p1 := plot([[seq(seq([i,j],i=-3..3),j=-3..3)]$3],style=point,
symbol=[circle,diamond,cross],color=black):
p2 := DEtools[DEplot](de,y(x),x=-3..3,y=-3..3,
dirgrid=[7,7],arrows=line,color=COLOR(RGB,.5,0,1)):
r1 := evalf(sqrt(5)):
p3 := plot([r1*cos(t),r1*sin(t),t=0..2*Pi],color=red):
t1 := plots[textplot]([[3.7,-.2,`x`],[-.3,3.7,`y`]]):
plots[display]([p1,p2,p3,t1],scaling=constrained); ;