Reduction of order for a third order differential equation

Question

Given $y'''(x) + P(x)y''(x) + Q(x)y'(x) + R(x)y(x) = 0$ with $P, Q$ and $R$ continuous functions and a solution $y_1(x)$ of the equation, how can we determine two other solutions. I used reduction of order to reduce this problem to a second order differential equation but I don't know how you can determine the other two solutions.

Answer 1

The second order equation could be essentially any second order homogeneous linear equation. It may well have no nontrivial closed form solutions, in which case you're out of luck.

For example, given any second order homogeneous linear equation $$ v'' + A(x) v' + B(x) v = 0$$ you can construct a third order equation $$ y''' + (A(x)-3) y'' + (3 - 2 A(x) + B) y' + (A(x) - B(x) - 1) y = 0$$ such that $y = e^x$ is one solution, and the other solutions are $y = e^x V$ where $v = V'$ satisfies $v'' + A(x) v' + B(x) v = 0$.