---
title: "Solving Initial Value Problems with Laplace Transforms"
description: "Transform the equation, solve the resulting algebra for the transform of the unknown, then invert. Initial conditions enter at the start rather than at the end, which is what makes the method suit dis"
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source: https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl.md
retrieved: 2026-09-12
---

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# Solving Initial Value Problems with Laplace Transforms

Transform the equation, solve the resulting algebra for the transform of the unknown, then invert. Initial conditions enter at the start rather than at the end, which is what makes the method suit discontinuous forcing.

Subject: Mathematics · Area: Differential Equations · Ages 21 to 22
Page: https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl

## Ready when they can

- Transform an initial value problem and solve for the transform of the solution
- Invert with partial fractions and a table
- Handle a forcing term that switches on at a fixed time

## Lesson: Trade calculus for algebra

The Laplace transform trades calculus for algebra. Each derivative becomes multiplication by s, with the starting values peeled off as constants. Your equation turns into one algebraic equation for Y, the transform of the answer.

**Example.** Take y double prime plus y equals zero, starting at one with zero slope. Transforming gives s squared Y minus s plus Y equals zero. Solving leaves Y as s over s squared plus one, whose table line is cosine of t.

Split hard transforms with partial fractions into first order chunks. Constants over s minus a invert to exponentials, and shifted quadratics invert to sines, cosines, and their damped cousins. Each chunk inverts by sight from the table.

Forcing that switches on at time a wears the step u of t minus a, which is zero before a and one from a onward. Match the argument first, then the transform gains the factor e to minus a s. A mismatched argument is the classic slip, so align before you transform.

**Recap.** Transform with the starting values inside, solve the algebra, split into table pieces, and shift delayed forcing with a matched step.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [The Laplace Transform](https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0)
- [Second-Order Linear Equations with Constant Coefficients](https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ)

## Opens up

- [Systems of Differential Equations and Eigenvalue Solutions](https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur)