> restart;with(DEtools):with(linalg):with(plots):
# An example of using Maple to find solutions to IVP for a given ODE
#  (Text, problem 4.8.37).
#  Consider the ODE (linear, homogeneous, constant coefficients):
> de1 := D(D(y))(x) - 4*D(y)(x) + 5*y(x) = exp(5*x)+x*sin(3*x)-cos(3*x);

           (2)
  de1 := (D   )(y)(x) - 4 D(y)(x) + 5 y(x) =

        exp(5 x) + x sin(3 x) - cos(3 x)

# The initial conditions are
> init_con := y(0) = 1, D(y)(0) = 0;

                  init_con := y(0) = 1, D(y)(0) = 0

# We now compute the solution using DSOLVE (if only interested in
# particular solution, neglect homogeneous part):
> solution := dsolve({de1, init_con},{y(x)});

                       957                   43
  solution := y(x) = - --- exp(2 x) sin(x) + -- exp(2 x) cos(x)
                       400                   50

                                                        13
         + 1/400 (16 + 30 x) cos(3 x) + 1/10 exp(5 x) + --- sin(3 x)
                                                        400

         - 1/40 x sin(3 x)

# Now some MAPLE esoterica: convert the FUNCTION y(x) to an EXPRESSION
# that can be evaluated and plotted;
# this is done with the command subs:
> expr := subs(solution,y(x));

            957                   43
  expr := - --- exp(2 x) sin(x) + -- exp(2 x) cos(x)
            400                   50

                                                        13
         + 1/400 (16 + 30 x) cos(3 x) + 1/10 exp(5 x) + --- sin(3 x)
                                                        400

         - 1/40 x sin(3 x)

> plot(expr,x=-20..1, axes=BOXED,title="solution to a 2nd order IVP");

>