# Number of linearly independent solutions for a second order linear inhomogeneous ODE and PDE

A second order, linear, homogeneous ordinary differential equation (ODE) has two linearly independent solutions.

$\bullet$ Is it also true for a second order, linear, inhomogeneous ODE?

$\bullet$ How many linearly independent solutions do a second order partial differential equation (PDE) have?

• In general the set of solutions of a nonlinear ODE is not a linear space. – Rigel May 20 '17 at 16:17
• A second order linear ODE in general has 3 linearly independent solutions, it solution space is a 2-dimensional affine space. Or did you speak about homogeneous linear ODE? – LutzL May 20 '17 at 16:36
• @LutzL Does linear homogeneous ODE and linear inhomogeneous ODE have different numbers of linearly independent solutions? Is there a proof of that? – SRS May 20 '17 at 17:01
• Any 3 points in generic position in a 2D plane that does not go through the origin are as vectors linearly independent. – LutzL May 20 '17 at 17:56

In the PDE case, it is more complicated. It likely depends on the equation you are considering. To take an example, in two dimensions the Laplace equation is a second order partial differential equation. In this case, the Cauchy Riemann equations give solutions to the Laplace equation. As you may know, there are an infinite number of solutions to these equations, namely the complex polynomials $z^n$ and their linear combinations. There are likely examples with finite numbers of solutions. So in this case it is likely that there is no general answer to your question.