= Linear partial differential equation
A linear partial differential equation is a <partial differential equation> in which the unknown function and its <partial derivatives> occur linearly, with coefficients independent of the unknown. Its homogeneous solutions obey the <superposition principle>.
For example, the <heat equation> permits adding independently decaying <Fourier modes>. For an equation $Lu=f$ with a fixed nonzero <PDE source term>, adding two solutions gives $L(u+v)=2f$, so the sum generally solves a different problem. Likewise, adding solutions adds their <initial conditions> and <boundary conditions>.
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