ODEs

1. Consider the ODE .
2. Find the unique solution of the ODE which satisfies the initial conditions

1. Plot the solution for  and notice the point on the graph.
1. Consider the ODE .
1. Solve the ODE and verify the general solution by substitution.
2. Find the unique solution that satisfies the initial conditions

1. Plot the solution for  and notice the point  on the graph.

Solutions

Following is the MAPLE sequence of steps to solve Example 1

> Example 1:#a]

> deq:=diff(y(x),x\$3)-3*diff(y(x),x\$2)-4*diff(y(x),x)+12*y(x)=sin(x);

> sol:=dsolve(deq,y(x));

> simplify(subs(y(x)=rhs(sol),deq));

> #b]

> sol2:=dsolve({deq,y(0)=0,D(y)(0)=-5,(D@@2)(y)(0)=10},y(x));

> #c]

> plot(rhs(sol2),x=-1..1,-6..11);

>

Following are the MAPLE steps to solve Example 2

> Example 2:#a]

> deq2:=t^2*diff(x(t),t\$2)+t*diff(x(t),t)+(t^2-1/4)*x(t)=t^(3/2);

> sol:=dsolve(deq2,x(t));

> simplify(subs(x(t)=rhs(sol),deq2));

> #b]

> sol2:=dsolve({deq2,x(5)=0,D(x)(5)=2},x(t));

> #c]

> plot(rhs(sol2),t=0..10,x=-2..4);

>

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