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 and initial conditions at t = 0.
  2. Take the Laplace transform of both sides.
  3. Use derivative transform rules for y′, y″, and higher derivatives.
  4. Substitute the initial conditions.
  5. Replace ℒ{y(t)} with Y(s).
  6. Solve for Y(s).
  7. Simplify or decompose Y(s) when useful.
  8. 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{y⁽ⁿ⁾} = sⁿY(s) − sⁿ⁻¹y(0) − ⋯ − y⁽ⁿ⁻¹⁾(0)
After solving for Y(s), y(t) = ℒ−1{Y(s)}

Example 1: y' + 2y = e^(-t), t, y, y(0)=1

Step 1 - Write the IVP.

In this problem: Start with the ODE and all required initial conditions at t = 0.

2·y(t) + Derivative(y(t), t) = exp(−t)
y(0) = 1

Step 2 - Take the Laplace transform of both sides.

In this problem: Apply ℒ{·} to the whole equation.

ℒ{2·y(t) + Derivative(y(t), t)} = ℒ{exp(−t)}

Step 3 - Use the derivative transform rules.

In this problem: Replace derivative transforms using initial-value Laplace formulas.

ℒ{y′} = sY(s) − y(0)

Step 4 - Write the transformed equation before substituting values.

In this problem: This shows the equation in Y(s) before the initial values are plugged in.

Y·s + 2·Y − y(0) = 1/(s + 1)

Step 5 - Substitute the initial conditions.

In this problem: Replace y(0), y′(0), and any higher initial values with the given numbers.

Y·s + 2·Y − 1 = 1/(s + 1)

Step 6 - Solve for Y(s).

In this problem: Collect the Y(s) terms and divide to isolate Y(s).

Y(s) = 1/(s + 1)

Step 7 - Simplify Y(s).

In this problem: A decomposed form makes the inverse Laplace step clearer when possible.

Y(s) = 1/(s + 1)

Step 8 - Take the inverse Laplace transform.

In this problem: Convert the expression in s back into a function of t.

y(t) = ℒ−1{1/(s + 1)}

Step 9 - Final answer.

In this problem: This is the solution that satisfies the ODE and initial conditions.

y(t) = exp(−t)

Final answer: See final step

Example 2: y'' + 3y' + 2y = sin(t), t, y, y(0)=0, y'(0)=1

Step 1 - Write the IVP.

In this problem: Start with the ODE and all required initial conditions at t = 0.

2·y(t) + 3·Derivative(y(t), t) + Derivative(y(t), (t, 2)) = sin(t)
y(0) = 0
y'(0) = 1

Step 2 - Take the Laplace transform of both sides.

In this problem: Apply ℒ{·} to the whole equation.

ℒ{2·y(t) + 3·Derivative(y(t), t) + Derivative(y(t), (t, 2))} = ℒ{sin(t)}

Step 3 - Use the derivative transform rules.

In this problem: Replace derivative transforms using initial-value Laplace formulas.

ℒ{y′} = sY(s) − y(0)
ℒ{y″} = s²Y(s) − s·y(0) − y′(0)

Step 4 - Write the transformed equation before substituting values.

In this problem: This shows the equation in Y(s) before the initial values are plugged in.

Y·s^2 + 3·Y·s + 2·Y − s·y(0) − 3·y(0) − Subs(Derivative(y(t), t), t, 0) = 1/(s^2 + 1)

Step 5 - Substitute the initial conditions.

In this problem: Replace y(0), y′(0), and any higher initial values with the given numbers.

Y·s^2 + 3·Y·s + 2·Y − 1 = 1/(s^2 + 1)

Step 6 - Solve for Y(s).

In this problem: Collect the Y(s) terms and divide to isolate Y(s).

Y(s) = (s^2 + 2)/(s^4 + 3·s^3 + 3·s^2 + 3·s + 2)

Step 7 - Simplify Y(s).

In this problem: A decomposed form makes the inverse Laplace step clearer when possible.

Y(s) = −(3·s − 1)/(10·(s^2 + 1)) − 6/(5·(s + 2)) + 3/(2·(s + 1))

Step 8 - Take the inverse Laplace transform.

In this problem: Convert the expression in s back into a function of t.

y(t) = ℒ−1{−(3·s − 1)/(10·(s^2 + 1)) − 6/(5·(s + 2)) + 3/(2·(s + 1))}

Step 9 - Final answer.

In this problem: This is the solution that satisfies the ODE and initial conditions.

y(t) = ((sin(t) − 3·cos(t))·exp(2·t) + 15·exp(t) − 12)·exp(−2·t)/10

Final answer: See final step