Basic Commands to Solve Differential Equations in Maple MTH 336 Differential Equations Lab by Karen Donnelly The following command clears Maple's internal memory -- as if starting over. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

To use Maple to solve a differential equation for us, it is best to first define the equation and then use Maple's dsolve command to solve. As a first example we consider a differential equation that we could readily solve ourselves: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW9HRiQ2LVEwJkRpZmZlcmVudGlhbEQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkgvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjQtRiM2Jy1JI21pR0YkNiNRIUYnLUYjNiZGMC1GUTYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnRktGNEZQRktGNC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXW8vJSliZXZlbGxlZEdGOS1GMTYtUTEmSW52aXNpYmxlVGltZXM7RidGNEY3RjpGPEY+RkBGQkZERkZGSS1GUTYlUSJ5RidGWUZmbi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUZRNiVRInhGJ0ZZRmZuRktGNEY0LUYxNi1RIj1GJ0Y0RjdGOkY8Rj5GQEZCRkQvRkdRLDAuMjc3Nzc3OGVtRicvRkpGZHAtSSVtc3VwR0YkNiVGXXAtSSNtbkdGJDYkUSIyRidGNC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGS0Y0. or y' = 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 We know the general solution to this is 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, where C is a constant. Enter the differential equation and save it in the variable named ODE: PkkkT0RFRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiRGLSokRi0iIiM= We can use the "short-hand" notation ' in Maple for the derivative as long as the independent variable is x. JSFH PkkkT0RFRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInhHRiRGLSokRi0iIiM= JSFH Ask Maple to solve the differential equation ODE for the function y(xa). LUknZHNvbHZlRzYiNiNJJE9ERUdGJA== Note that Maple uses _C1 to denote the arbitrary constant in the solution. Also note that in both commands we since x is the independent variable and y the dependent variable, we use the functional notation y(x). To get a unique solution we need an initial condition, say 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. Define the initial condition for the differential equation and save in the variable names IC: PkkjSUNHNiIvLUkieUdGJDYjIiIhIiIi Now solve the differential equation with the specified initial conditions: LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiM8JEkjSUNHRidJJE9ERUdGJw== JSFH

We now consider a second order differential equation: 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 Enter the differential equation and save it in the variable named ODE: PkkkT0RFRzYiLywoLUklZGlmZkclKnByb3RlY3RlZEc2JS1JInlHRiQ2I0kieEdGJEYuRi4iIiItRig2JEYrRi4iIiNGKyIjPCIiIQ== Simplified notation: PkkkT0RFRzYiLywoLUklZGlmZkclKnByb3RlY3RlZEc2JS1JInlHRiQ2I0kieEdGJEYuRi4iIiItRig2JEYrRi4iIiNGKyIjPCIiIQ== Ask Maple to solve the differential equation ODE for the function y(x). LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJE9ERUdGJw== Note since this is a second order differential equation we get a two parameter family of solutions. Hence Maple provides two arbitrary constants _C1 and _C2. To get a unique solution we must provide two initial conditions, one for y and one for its first derivative. Assume the following initial conditions: 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, y'(0) = -1 Define the initial conditions for the differential equation and save in the variable names IC. We can Maple's differential operator D for the first derivative or the "'" notation. D(y) computes the first derivative of y as a function. Thus D(y) (0) is the derivative of y evaluated at 0. Alternatively, we can use the shorcut notation y'(0) to achieve the same. PkkjSUNHNiI2JC8tSSJ5R0YkNiMiIiEiIiIvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRikhIiI= Solve the differential equation with the specified initial conditions: Pkkkc29sRzYiLUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiM8JEkjSUNHRiRJJE9ERUdGJA== Checking the solution: We can check this solution by hand to see that it satisfies both the differential equation and the two initial conditions. We can use a little Maple assistance also with this. The sytax is a little awkward. First we use the unapply operator to the right hand side of the solution. This is used define a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy whose value at t is the right hand side of the equation representing the solution. LUkkcmhzRyUqcHJvdGVjdGVkRzYjSSRzb2xHNiI= PkkkcGhpRzYiLUkodW5hcHBseUdGJDYkLUkkcmhzRyUqcHJvdGVjdGVkRzYjSSRzb2xHRiRJInhHRiQ= QyUtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkkcGhpRzYiNiNJInhHRilGKyIiIi1GJDYlRidGK0Yr We now ask Maple to compute the first and second derivatives of the proposed solution: QyUtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkkcGhpRzYiNiNJInhHRilGKyIiIi1GJDYlRidGK0Yr We could now substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy and its derivatives into the left hand side of differential equation by hand -- but we ask Maple instead to do this for us. We see that it does simplify to 0. LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYlLUkkcGhpRzYiNiNJInhHRilGK0YrIiIiLUYkNiRGJ0YrIiIjRiciIzw= Verifying the initial conditions: QyUtSSRwaGlHNiI2IyIiISIiIi1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmR0YrNiQtRiQ2I0kieEdGJUYyL0YyRic=

Exercise 1: a) Use Maple to define and solve the differential equation -- (Note: use the expression pallete for the exponential (do not just type in "e" -- it won't work!). 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= b) Now solve the same equation with the initial condition 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Exercise 2: a) Use Maple to define and solve the differential equation 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= b) Now solve the same equation with the initial conditions y(0) = 0, y'(0) = 0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=