Maple can solve some higher-order ODEs explicitly:

> dsolve( { (D@@2)(y)(t) = 4 - sin(t) + y(t) - 2*D(y)(t), y(0) = 1, D(y)(0) = 2 } );

Here, the derivative of `y(t)` is represented by
`D(y)(t)` and the second derivative by
`(D@@2)(y)(t)` (applying the derivative twice).

The second example cannot be solved by Maple, as:

> dsolve( { (D@@3)(y)(t) = y(t) - t*D(y)(t) + 4*(D@@2)(y)(t), y(0) = 1, D(y)(0) = 2, (D@@2)(y)(0) = 3 } );

returns no solution. However, you can ask for a numeric answer:

> dsolve( { (D@@3)(y)(t) = y(t) - t*D(y)(t) + 4*(D@@2)(y)(t), y(0) = 1, D(y)(0) = 2, (D@@2)(y)(0) = 3 }, 'numeric' ); proc(x_rkf45) ... end proc

From the name of the variable, you should be able to guess which method is being used to find numeric solutions to the given ODE.

For interest, Figure 1 shows a plot of the solution to the second IVP.

Figure 1. The solution to the 3rd-order IVP.

If we look at the initial conditions, this suggests that:

- the value of the solution at 0 should be 1,
- the value of the derivative at 0 should be 2, and
- the concavity at 0 should be 3.

Consequently, the solution should look like the parabola with similar properties, namely

^{2}+ 2 t + 1

If we plot both the solution and the parabola, we see in Figure 2 that, at 0, there is a reasonable match.

Figure 2. The solution and a fitting parabola.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.