# How To Solve Differential Equations Using Laplace Transform

**How To Solve Differential Equations Using Laplace Transform**. You can use the laplace transform to solve differential equations with initial conditions. Since you know how to solve it using lt (i do not think it is possible to do so, but solving it using an integrating factor or as an exact equation is doable), proceed and solve it that way while maintaining a constant for y ( 0) = c in all the calculations.

We’re just going to work an example to illustrate how laplace transforms can. In this section we will work a quick example using laplace transforms to solve a differential equation on a 3rd order differential equation just to say that we looked at one with order higher than 2nd. Use laplace transform to solve the differential equation y ″ − y ′ − 2y = sin(3t) with the initial conditions and y ′ (0) = − 1.

### Simply Take The Laplace Transform Of The Differential Equation In Question, Solve That Equation Algebraically, And Try To Find The Inverse Transform.

Equation for example 1 (a): The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. At this time, i do not offer pdf’s.

### Yl > E T @ Dt Dy 3 2 » ¼ º

T x ″ − t x ′ + 4 x = 2 e t, x ( 0) = sinh ( 1); Here are a set of practice problems for the laplace transforms chapter of the differential equations notes. Take the laplace transform of both sides.

### As We Will See In Later Sections We Can Use Laplace Transforms To Reduce A Differential Equation To An Algebra Problem.

Let y(s) be the laplace transform of y(t) take the laplace transform of both sides of the given differential equation. L{y ″ − y ′ − 2y} = l{sin(3t)} This video shows how to solve differential equations using laplace transforms.

### Once You Have The Solution For Y.

As we’ll see, outside of needing a formula for the laplace transform of y''', which we can get from the general formula, there is no real difference in how laplace transforms are used for. Use laplace transform to solve the differential equation y ″ − y ′ − 2y = sin(3t) with the initial conditions and y ′ (0) = − 1. Since you know how to solve it using lt (i do not think it is possible to do so, but solving it using an integrating factor or as an exact equation is doable), proceed and solve it that way while maintaining a constant for y ( 0) = c in all the calculations.

### Using The Transform Of A Derivative We Get:

The laplace solves de from time t = 0 to infinity. Solve the transformed system of algebraic equations for x,y, etc. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section.