Mr. Integral
Math notes/Differential Equations/First-Order Methods

First-Order Linear Equations — Cheat sheet

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Core rule

μ=e∫p dt,\mu=e^{\int p\,dt}, (μy)′=μq.(\mu y)'=\mu q.

Watch for

The complementary part carries initial-condition memory; the forced part reflects input. Their long-time behavior depends on the coefficient sign. A positive constant damping coefficient erases initial differences exponentially.