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  "body": "\n## Summary\n\nThe Mikusiński operational calculus, created by Jan Mikusiński, is an algebraic foundation of [[operational calculus|8050R3Z6HA]] that does not use integral transforms.\nFunctions on $[0, \\infty)$ are elements of the [[Mikusiński field of operators|82YAXZMMQ1]], in which differentiation is multiplication by an element $s$ up to initial values.\nLinear differential equations with constant coefficients become linear equations in this field, and the solution is read off by algebra.\nUnlike the Laplace transform approach, it needs no growth condition on the functions.\n\n## Explanations\n\n### L1 human\n\nMikusiński replaced the Laplace transform by pure algebra.\nHe made fractions of functions, with convolution as multiplication, much as one makes fractions of whole numbers.\nIn this world the derivative is multiplication by a fixed element $s$, corrected by the initial value.\nSolving $y' = a y$, $y(0) = 1$ becomes $s y - 1 = a y$, so $y = 1/(s - a)$, and one checks that this fraction is the function $e^{at}$.\n",
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