Laplace Transform ODE Solver Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Pro

This Laplace Transform ODE Solver Calculator solves an initial value problem by moving it out of the time domain, and shows the working line by line. Transforming both sides turns each derivative into a multiple of s, so the initial conditions are built into the algebra rather than fitted at the end. Solving for the transform and inverting it returns the answer in t. Following the same pattern every time keeps the initial conditions from being forgotten.

Step-by-step method

  1. Set up the problem.
  2. Take the Laplace transform of both sides using the derivative rules.
  3. Substitute the initial conditions and solve for Y(s).
  4. Take the inverse Laplace transform to get y(t).

Formula:

\(\mathcal{L}\{y'\} = sY(s) - y(0), \quad \mathcal{L}\{y''\} = s^{2}Y(s) - s\,y(0) - y'(0)\)
See Example 1 Hide Example 1

Problem

\(- y{\left(t \right)} + \frac{d^{2}}{d t^{2}} y{\left(t \right)} = 0\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Take the Laplace transform of both sides using the derivative rules.
  3. Substitute the initial conditions and solve for Y(s).
  4. Take the inverse Laplace transform to get y(t).

Step 1

Step 1 - Set up the problem.

In this problem: We solve \(- y{\left(t \right)} + \frac{d^{2}}{d t^{2}} y{\left(t \right)} = 0\) with \(y(0) = 0,\; y'(0) = 1\).

\(- y{\left(t \right)} + \frac{d^{2}}{d t^{2}} y{\left(t \right)} = 0,\quad y(0) = 0,\; y'(0) = 1\)

Step 2

Step 2 - Take the Laplace transform of both sides using the derivative rules.

In this problem: Transform each derivative using the Laplace derivative rules.

\(\mathcal{L}\{y'\} = sY(s) - y(0), \quad \mathcal{L}\{y''\} = s^{2}Y(s) - s\,y(0) - y'(0)\)

Step 3

Step 3 - Substitute the initial conditions and solve for Y(s).

In this problem: Substitute the initial conditions and solve the algebraic equation for Y(s).

\(y(0) = 0,\; y'(0) = 1\)

Step 4

Step 4 - Take the inverse Laplace transform to get y(t).

In this problem: Taking the inverse transform gives \(y(t) = \sinh{\left(t \right)}\).

\(y(t) = \sinh{\left(t \right)}\)

Final Answer

\(y(t) = \sinh{\left(t \right)}\)
See Example 2 Hide Example 2

Problem

\(- 3 y{\left(t \right)} + \frac{d}{d t} y{\left(t \right)} = 0\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Take the Laplace transform of both sides using the derivative rules.
  3. Substitute the initial conditions and solve for Y(s).
  4. Take the inverse Laplace transform to get y(t).

Step 1

Step 1 - Set up the problem.

In this problem: We solve \(- 3 y{\left(t \right)} + \frac{d}{d t} y{\left(t \right)} = 0\) with \(y(0) = 1\).

\(- 3 y{\left(t \right)} + \frac{d}{d t} y{\left(t \right)} = 0,\quad y(0) = 1\)

Step 2

Step 2 - Take the Laplace transform of both sides using the derivative rules.

In this problem: Transform each derivative using the Laplace derivative rules.

\(\mathcal{L}\{y'\} = sY(s) - y(0), \quad \mathcal{L}\{y''\} = s^{2}Y(s) - s\,y(0) - y'(0)\)

Step 3

Step 3 - Substitute the initial conditions and solve for Y(s).

In this problem: Substitute the initial conditions and solve the algebraic equation for Y(s).

\(y(0) = 1\)

Step 4

Step 4 - Take the inverse Laplace transform to get y(t).

In this problem: Taking the inverse transform gives \(y(t) = e^{3 t}\).

\(y(t) = e^{3 t}\)

Final Answer

\(y(t) = e^{3 t}\)

Example Problems

Click one to load it into the box and solve it.