Calculators / Differential Equations / Laplace Transform ODE Solver

Laplace Transform ODE Solver

Published on: December 28, 2026

This Laplace Transform ODE Solver helps you solve differential equations using Laplace transforms and shows each step clearly. It works by transforming the differential equation into an algebraic equation in the Laplace domain, applying the given initial conditions, and then using the inverse Laplace transform to return to the original function. This makes it useful for checking answers, understanding how the Laplace transform method works, and practising differential equations step by step.

Step-by-step method

  1. Write the ODE with initial conditions at t = 0.
  2. Take the Laplace transform of both sides.
  3. Substitute initial conditions and simplify to an equation in Y(s).
  4. Solve for Y(s).
  5. Take the inverse Laplace transform to get y(t).

Formula bank:

Laplace: ℒ{f(t)} = F(s)
ℒ{y′} = sY(s) − y(0)
ℒ{y″} = s²Y(s) − s·y(0) − y′(0)
ℒ{1} = 1/s
ℒ{e^{at}} = 1/(s − a)
ℒ{sin(bt)} = b/(s² + b²)
ℒ{cos(bt)} = s/(s² + b²)