Problem

Find a solution to the 4th-order differential equation

What can you say about

Solution

Start with y'''' = cos(x). Take the antiderivative of each side, to get

Continue taking antiderivatives until we reach y on the left hand side.

This is the general form of the solution. A particular solution would be found by specifying conditions to determine the constants C, D, E, and F.

Since the 4th derivative of cos(x) is cos(x), taking the derivative 4 more times would give us cos(x) again. So we can say that the solution to the 8th-order differential equation

equals

y = cos(x) + a degree 7 polynomial in x.