Partial Fractions & Laplace Workbench
Rational functions decomposed with the coefficient system shown, and the Laplace transform table parameterized in both directions — the pairing every ODE solution runs on. Covers §5.18 and §5.38–5.39.
Decompose P(x) / [(x−r₁)(x−r₂)(x−r₃)…]
Numerator up to degree 3; denominator as 2–4 distinct linear factors. Degree of P must be less than the number of factors (proper fraction).
Transform Table — Parameterized Both Ways
Linearity Builder
ℒ is linear: the transform of c₁f₁ + c₂f₂ is c₁F₁ + c₂F₂. Combine two table entries:
Inverse Laplace in practice = partial fractions first: split F(s) into table-shaped pieces, then read each piece backwards. That is exactly why these two tools share a page.