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  "body": "\n## Summary\n\nA complete integral of a first-order [[partial differential equation|GC5BDZ6QBC]] in $n$ variables is a family of solutions depending on $n$ independent parameters.\nFrom such a family one builds many more solutions by taking envelopes of subfamilies.\nComplete integrals are the classical route to solving [[Hamilton–Jacobi equations|5TRXQKP7DY]] in mechanics and Clairaut-type equations.\nSee the [[definition|KEC5TGS9WM]].\n\n## Explanations\n\n### L1 human\n\nFor a first-order equation in $n$ variables, one often finds a simple family of solutions, such as planes with a free slope.\nIf the family has $n$ genuinely different parameters, it is called a complete integral.\nFor example, every plane $u = a_1 x_1 + a_2 x_2 + a_1^2 + a_2^2$ solves the equation $u = x \\cdot \\nabla u + \\abs{\\nabla u}^2$.\nBending these planes together, by taking the envelope of a one-parameter subfamily, produces new curved solutions.\n",
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