Using Laplace transform to solve ODE

• Rumpelstiltzkin
In summary, the solution to the given differential equation is y = \frac{sint}{t}, with the initial condition y(0) = 0.
Rumpelstiltzkin
Here's the problem:

Solve $$ty'' + 2y' + ty = 0$$
with $$lim_{t\rightarrow0+} y = 0$$
and
$$y(\pi) = 0$$

So here's my working:

$$-\frac{d}{ds}[s^2Y - sy(0) - y'(0)] + 2[sY - y(0)] - Y' = 0$$

$$-[2sY + s^2Y' - s] + 2[sY - y(0)] - Y' = 0$$

Arranging terms, I get:

$$Y' = \frac{s-2y(0)}{s^2+1}$$
and
$$-Y' = \frac{2y(0)-s}{s^2+1}$$

$$L^{-1}(-Y')= L^{-1}(\frac{2y(0)}{s^2+1}) - L^{-1}(\frac{s}{s^2+1})$$

$$ty = 2y(0)sint - cost$$

$$y = \frac{2y(0)sint}{t} - \frac{cost}{t}$$
The answer is $$y = \frac{sint}{t}$$

What's wrong?

Thanks..

Last edited:
The problem is that you have not included the initial condition y(0) in your solution. The general solution of the differential equation is y = \frac{2y(0)sint}{t} - \frac{cost}{t}, but since y(0) = 0, the solution simplifies to y = \frac{sint}{t}.

1. How does Laplace transform help in solving Ordinary Differential Equations (ODEs)?

Laplace transform converts a differential equation into an algebraic equation, making it easier to solve. This is because it transforms the derivatives in the equation into algebraic expressions, which can then be manipulated using algebraic methods.

2. What types of ODEs can be solved using Laplace transform?

Laplace transform can be used to solve linear, constant coefficient, and initial value problems for ODEs. It is also useful for solving ODEs with discontinuous or piecewise continuous functions.

3. What are the steps involved in using Laplace transform to solve an ODE?

The steps involved in using Laplace transform to solve an ODE are: 1) Taking the Laplace transform of both sides of the equation, 2) Applying algebraic operations to solve for the transformed function, 3) Inverse transforming the solution to obtain the solution in the original domain, and 4) Checking the solution for accuracy.

4. Are there any limitations to using Laplace transform for ODEs?

There are certain limitations to using Laplace transform for ODEs. It is not suitable for solving ODEs with non-constant coefficients or with variable coefficients. It also cannot be used for solving ODEs with variable initial conditions or boundary conditions.

5. Can Laplace transform be used for solving partial differential equations (PDEs)?

Yes, Laplace transform can be used to solve certain types of PDEs, such as linear, constant coefficient, and initial value problems. However, it is not as commonly used for PDEs as it is for ODEs, and there are other methods that may be more suitable for solving PDEs.

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