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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?

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    $\begingroup$ In general the set of solutions of a nonlinear ODE is not a linear space. $\endgroup$ – Rigel May 20 '17 at 16:17
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    $\begingroup$ 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? $\endgroup$ – LutzL May 20 '17 at 16:36
  • $\begingroup$ @LutzL Does linear homogeneous ODE and linear inhomogeneous ODE have different numbers of linearly independent solutions? Is there a proof of that? $\endgroup$ – SRS May 20 '17 at 17:01
  • $\begingroup$ Any 3 points in generic position in a 2D plane that does not go through the origin are as vectors linearly independent. $\endgroup$ – LutzL May 20 '17 at 17:56
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For the first question, an inhomogeneous equation should have the same number of linearly independent solutions as the homogeneous equation. This is because if you have a particular solution to the inhomogeneous equation, you can add solutions to the homogeneous equation and still have solutions. Conversely, if you have two inhomogeneous solutions, their difference is homogeneous. Therefore you can map the inhomogeneous solutions to homogeneous solutions, so they are effectively in one to one correspondence. (This assumes that a solution exists, there are likely pathological examples that break this assumption, in which case there would be no solutions.)

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.

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