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  "body": "\n## Summary\n\nSeparation of variables is a method for solving differential equations by splitting the dependence on different variables.\nFor an ordinary differential equation $y' = g(x)\\, h(y)$, one moves the $y$ terms to one side and the $x$ terms to the other and integrates.\nSuch equations are called [[separable equations|074E9A8ZJK]].\nFor a linear partial differential equation, one looks for solutions that are products of functions of one variable each.\nThis turns the equation into several ordinary differential equations.\nIt is one of the oldest and most used solution methods.\nSee the [[definition|PR2DV03M4H]].\n\n## Explanations\n\n### L1 human\n\nThe idea is to put everything about one variable on one side of the equation and everything about the other variable on the other side.\nFor $y' = x y$, one writes $dy / y = x\\, dx$, integrates both sides to get $\\ln\\abs{y} = x^2/2 + C$, and finds $y = A e^{x^2/2}$.\nFor the heat equation $u_t = u_{xx}$, one guesses $u(x, t) = X(x)\\, T(t)$, and the equation splits into one equation for $X$ and one for $T$.\nSums of such product solutions then give many more solutions, which leads to Fourier series.\n",
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