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# Solving Initial Value Problems : Laplace Transforms and Convolution Theorem

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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)

https://brainmass.com/math/calculus-and-analysis/initial-value-problems-laplace-transforms-convolution-theorem-117986

#### 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.

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.

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