ODE's with MapleWe illustrate Maple's differential equation solving ability by looking at an example that gives us an explicit solution, y'=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, y(1)=0. Notice how we enter the differential equation. Since y is the dependent variable and t the independent variable, we must always use y(t) instead of just y.restart;deq:=diff(y(t),t)=(2/t)*y(t)+t^2*exp(t);We use the command infolevel to determine the level that Maple shows us about its approach to solving a problem. The default level is 1, 5 the maximum level.infolevel[dsolve]:=5;We use dsolve to find the general solution to the differential equation.sol:=dsolve(deq,y(t));We can use odetest to test the correctness of our solution. For a correct solution, 0 is returned, although sometimes we may need to do a simplification to get 0.odetest(%,deq);We enter the initial condition and find the particular solution.init:=y(1)=0;deqsol:=dsolve({deq,init},y(t));odetest(%,deq);After setting infolevel back to 1, we load the DEtools package, which contains functions that help you work with differential equations.infolevel[dsolve]:=1;with(DEtools);Let's look at the direction field for our differential equation. Each arrow shows the direction of movement from a particular point in the plane and has slope 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.DEplot (deq,y(t),t=1..2,y=-10..10,color=magenta,arrows = medium);We superimpose the graph of the solution curve on the direction field.DEplot (deq,y(t),t=1..2,y=-10..10,{[1,0]},linecolor=green,color=magenta,arrows = medium);We do the following to find the solution at a particular value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, such as t=1.5.q:=rhs(deqsol);"y(1.5)"=evalf(eval(q,t=1.5));We can also do this by changing the right hand side into a function.y:=unapply(q,t);
"y(1.5)"=evalf(y(1.5));
"y(2)"=evalf(y(2));;Now let's look at the solutions corresponding to the initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIxRidGQEZARkBGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRl1vLUYsNiVRImlGJ0Y2RjlGQEYrRkA=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = -5..5. Let's superimpose these solutions on the direction field.y:='y';DEplot (deq,[y(t)],t=1..2,y=-10..10,{[1,-5],[1,-4],[1,-3],[1,-2],[1,-1],[1,0],[1,1],[1,2],[1,3],[1,4],[1,5]},linecolor=green,color=magenta,arrows = medium);Some differential equations cannot be solved by Maple. This is one reason why we have need of numerical methods. We look at 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.infolevel[dsolve]:=5;deq2:=D(y)(t)=1+t*sin(t*y(t));deqsol2:=dsolve(deq2,y(t));We see that Maple cannot find an exact solution. Let's look at the direction field.infolevel[dsolve]:=1;DEplot (deq2,y(t),t=-5..5,y=-5..5,color=magenta,arrows = medium);Kind of wild!