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Let $L$, $M$ and $N$ be linear differential operators with $N=L+M$.

True or false: If the function $u$ solves $L[u]=f$ and the function $v$ solves $M[v]=g$, does this necessarily imply that the function $w=u+v$ solves $N[w]=f+g$ ?

My approach: $N[w]=L[w]+M[w]=L[u+v]+M[u+v]=L[u]+L[v]+M[u]+M[v]=f+g+L[v]+M[u]$

I would therefore say that the answer is false because the proposed statement is true if and only if $M[u]=0$ AND $L[v]=o$, i.e. if and only if $u$ and $v$ solve the two PDEs.

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    Right you are! The functions $u$ and $v$ must solve both equations. – 2017-01-31

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