Are PDEs that satisfy the principle of superposition: if $u$ and $v$ are solutions then any linear combination $au+bv$ is also a solution. In other words, the solution space (by the way, check solution space of an ODE) is a vector space. If the space of solutions is an affine space then it is a inhomogeneous linear PDE. In any other case it is a nonlinear PDE.
They can be expressed in the form $\mathcal{P}[u(x)]=0$, where $\mathcal{P}$ is a linear differential operator.
When the coefficients of this differential operator are constant it has translation invariance: $\mathcal{P}[u(x-a)]=\mathcal{P}u$. From here can be shown that solutions of these equations can be translated and still remain solutions.
Related system of linear first order homogeneous PDEs
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Author of the notes: Antonio J. Pan-Collantes
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