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  "body": "\n## Summary\n\nThe Laplace transform method solves a [[Volterra equation of convolution type|SNFYTC1TTA]] on $[0, \\infty)$ by turning it into an algebraic equation.\nThe [[convolution theorem for the Laplace transform|0JNMJEDD93]] changes the convolution $k * x$ into the product $\\hat k \\, \\hat x$ of the transforms.\nOne solves for $\\hat x$ and then inverts the transform.\nIt is the standard way to find explicit solutions, and also to study the long-time behaviour of solutions.\n\n## Explanations\n\n### L1 human\n\nA convolution equation mixes the unknown with its own past, which makes it hard to solve directly.\nAfter a [[Laplace transform|0V49JEZW06]], the mixing becomes a plain multiplication.\nFor example, $x(t) = 1 + \\int_0^t x(s) \\, ds$ becomes $\\hat x(p) = \\frac1p + \\frac1p \\hat x(p)$.\nSo $\\hat x(p) = \\frac{1}{p - 1}$, which is the transform of $e^t$, and indeed $x(t) = e^t$.\n",
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