What can we say about the number of linearly independent solutions to a PDE?
Firstly, in what function space are we talking about linearity?
Secondly, if there is not a general conclusion, what if we restrict the question to a linear PDE?
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Google didn't tell you anything?...;) – Zach466920 Jan 09 '16 at 16:32
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There are linear PDEs with no non-zero solutions. Others have infinitely many solutions. With non-singular linear ODEs of order $N$ on an interval $[a,b]$, you have a solution space with dimensiion equal to the number of endpoint conditions you can specify at $a$ such as $f(a)=A_1,f'(a)=A_2,\cdots,f^{(N-1)}(a)=A_N$. For a PDE such as Laplace's equation $\nabla^2f=0$ on a bounded region with a smooth boundary, you can specify function values on a closed boundary, which leads to an infinite-dimensional solution space. – Disintegrating By Parts Jan 09 '16 at 19:00