# Solving Initial Value Problems : Laplace Transforms and Convolution Theorem

1. For the problem given below use the convolution theorem to write a formula for the solution of the I.V. problem in terms of f(t)

y''-5y'+6y=f(t)

y(0) = y'(0)=0

2. Use Laplace Transforms to solve the following equation

t^2 y'-2y = 2 (no IC's)

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#### Solution Preview

I am still puzzled regarding the second one. Why on earth one would want to demonstrate the Laplace transform method on a problem which only complicates the solution? Anyway, in the end I solve it again, this time in a more direct way, just to show how to get to the solution without all the transform stuff. It was also a good check that I actually got the correct result.

Please see the attached file.

To solve the differential equation with Laplace transform will use the following attributes of the transform and its inverse.

The transform is linear:

The transform is invertible:

The convolution property:

A convolution between two functions is defined ...

#### Solution Summary

An IVP is solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.