# Solving Initial Value Problem with Laplace Transform

• Kleanthis
In summary, the conversation is about solving an initial value problem using Laplace transform. The given equation is y''(x)-xy'(x)+y(x)=5 with initial values y(0)=5 and y'(0)=3. The equation is transformed into sY'(s)+(s^2+2)Y(s)=5/s^2 + 5s + 3, which is a nonlinear first order differential equation with the variable s. There is a disagreement about the presence of Y'(s) in the equation, but it is clarified that the factor xy'(x) is responsible for it. The conversation also touches on the limitations and preferences of using Laplace transform, as well as the importance of specifying all necessary
Kleanthis
Hello from Greece!Gongrats for your forum.

I was wondering if anyone could give me a hand with this initial value problem.
It s to be solved via Laplace transform.

y''(x)-xy'(x)+y(x)=5 , y(0)=5 and y'(0)=3

Applying the transform to the given equation I end up to :

sY'(s)+(s^2+2)Y(s)=5/s^2 + 5s + 3

This is a non linear first order differential equation with the variable s.

Any ideas to continue?

How do you have Y'(s) in there? Laplace transform takes DE and turns it into an algebraic equation. It takes PDEs and turns them into DEs.

Check the given equation again to see why is that Y'(s) over there.The equation I end up is correct.

Kleanthis said:
Check the given equation again to see why is that Y'(s) over there.The equation I end up is correct.

From http://en.wikipedia.org/wiki/Laplace_transform

"It is commonly used to produce an easily solvable algebraic equation from an ordinary differential equation ."

The factor xy'(x) gives you Y'(s) when you apply the Laplace transform.If it s not clear enough I ll write it analytical.

I really dislike the Laplace transform method! It only works on linear equations with constant coefficients and there are much easier ways of solving such problems.

By the way, the orginal problem was given as
y''(t)+ 4y'(t) = sin2t

y(0) = 0

Did no one point out that that is a second order equation and just saying "y(0)= 0" is not enough to specify the solutions?
That has an infinite number of solutions with different values for y' at t= 0.

Kleanthis said:
The factor xy'(x) gives you Y'(s) when you apply the Laplace transform.If it s not clear enough I ll write it analytical.

You are right, let me think about it some more.

## 1. What is a Laplace transform and its significance in solving initial value problems?

A Laplace transform is a mathematical tool that allows us to solve differential equations in a simpler algebraic form. It transforms a function of time into a function of complex frequency, making it easier to solve initial value problems (IVPs) using algebraic methods rather than differential equations. It is particularly useful for solving linear IVPs with constant coefficients.

## 2. How does the Laplace transform method work in solving IVPs?

The Laplace transform method works by transforming the given differential equation and initial conditions into an algebraic equation in the complex frequency domain. The problem is then solved using algebraic techniques, and the solution is obtained by taking the inverse Laplace transform of the solution in the frequency domain. This method is especially useful for solving higher order linear differential equations.

## 3. What are the advantages of using Laplace transform in solving IVPs?

There are several advantages of using the Laplace transform method in solving IVPs. Firstly, it simplifies the process of solving differential equations by converting them into algebraic equations. It also eliminates the need for integration, which can be a tedious and error-prone process. Additionally, the Laplace transform method can be used to solve a wide range of differential equations, including those with discontinuous or non-constant coefficients.

## 4. Are there any limitations of using the Laplace transform method for solving IVPs?

While the Laplace transform method is a powerful tool for solving IVPs, it does have some limitations. It can only be applied to linear differential equations and is not applicable to nonlinear or partial differential equations. Additionally, the initial conditions must be specified at t=0 for the method to work effectively.

## 5. Can Laplace transform be used to solve IVPs with variable coefficients?

Yes, the Laplace transform method can be used to solve IVPs with variable coefficients. However, the process can be more complex and may require additional techniques, such as partial fraction decomposition, to obtain the final solution. It is important to note that the Laplace transform method is most effective for solving IVPs with constant coefficients, but with the right approach, it can also be applied to problems with variable coefficients.

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