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annas425

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**Laplace transform initial value problem--need help!**

Looking at the solutions to these initial value problems, I am very confused as to how the highlighted steps are derived (both use heaviside step functions). I know the goal is to get the fractions in a familiar form so that one can look them up in a "Common Laplace Transform" Table to take the inverse and get the solution. But did they do this using partial fraction expansion? Or some other Algebra tricks that I am unaware of? Any clarity or help with either of these problems is MUCH appreciated!

First Problem:

Determine the solution to x' - 5x = f(t), where x(0)=0 and f(t) = 0 for t<3; f(t) = t for 3<=t<4; f(t) = 0 for t>=4.

Solution...

Second Problem:

Determine the solution to x'' + 25x = f(t), where x(0)=0 and x'(0)=0 and f(t) = t for 0<=t<1; f(t)= cos(t-1) for t>=1.

Solution...

Thank you so much, in advance! I don't need someone to entirely work out these problems; I just need some light shed on the highlighted steps!