Laplace Transform ODE Solver Calculator

This Laplace Transform ODE Solver solves initial value problems such as y'' + y = 0 with y(0) and y'(0) given. It transforms the equation, solves for Y(s), and inverts to get y(t).

Step-by-step method

1. Write the ODE and initial conditions.
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:

Example 1:

Step 1 - Write the ODE and initial conditions.

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

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

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

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).

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

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

Final answer:

Example 2:

Step 1 - Write the ODE and initial conditions.

In this problem: We solve \(2 y{\left(t \right)} + \frac{d}{d t} y{\left(t \right)} = 0\) with \(y(0) = 3\).

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

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

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).

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

In this problem: Taking the inverse transform gives \(y(t) = 3 e^{- 2 t}\).

Final answer:

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. Write the ODE and initial conditions.
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 - Write the ODE and initial conditions.

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 \(y(t) = \sin{\left(t \right)}\).

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

Final Answer

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

See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

1. Write the ODE and initial conditions.
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 - Write the ODE and initial conditions.

In this problem: We solve \(2 y{\left(t \right)} + \frac{d}{d t} y{\left(t \right)} = 0\) with \(y(0) = 3\).

\(2 y{\left(t \right)} + \frac{d}{d t} y{\left(t \right)} = 0,\quad y(0) = 3\)

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).

\(y(0) = 3\)

Step 4

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

In this problem: Taking the inverse transform gives \(y(t) = 3 e^{- 2 t}\).

\(y(t) = 3 e^{- 2 t}\)

Final Answer

\(y(t) = 3 e^{- 2 t}\)