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  "body": "\n## Summary\n\nA linear PDE is a [[partial differential equation|GC5BDZ6QBC]] in which the unknown and its derivatives appear only linearly.\nIts coefficients depend on the variables alone.\nFor a linear homogeneous equation, sums and multiples of solutions are again solutions.\nThis superposition principle makes linear PDEs the best understood class, and it is the setting of Fourier methods, fundamental solutions and distribution theory.\nSee the [[definition|B54GEJWQAV]].\n\n## Explanations\n\n### L1 human\n\nIn a linear PDE each term is a known function times one derivative of the unknown $u$, plus possibly a known source term.\nNo products such as $u \\cdot \\partial_x u$ and no powers such as $u^2$ occur.\nThe [[heat equation|VP2HN86VAG]] $\\partial_t u = \\partial_x^2 u$ is linear: if $u$ and $v$ solve it, so does $2u + 3v$.\nThe equation $\\partial_t u + u \\, \\partial_x u = 0$ is not linear, and the sum of two of its solutions is usually not a solution.\n",
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