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I know that with second order linear differential equations of the form $y''+p(x)y'+q(x)y=0$ if you have solutions $y_1$ and $y_2$, then $Ay_1+By_2$ for $A,B\in \mathbb{R}$ is also a solution because $$Ay''_1+By''_2+p(x)(Ay'_1+By'_2)+q(x)(Ay_1+By_2)$$ $$=A(y''_1+py'_1+qy_1)+B(y''_2+py'_2+qy_2)$$ $$=A(0)+B(0)=0$$ So I basically want to know if there are other types of differential equations with this property.

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Saying that you can write all solutions as linear combinations of "n" specific solutions is saying that the set of solutions form a vector space and that the differential operator acting on then is a linear operator. So, no, there are no other equations that have that property- that property is pretty much the definition of "linear homogeneous" differential equation.

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Linear differential equations in general (of arbitrary degree) have this property, for basically the same reason.

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