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Solve Differential Equations Using Laplace Transforms 

If you haven't seen Laplace Transforms before, the calculus section of 17calculus has a complete discussion of what they are and how to work with them. 
Using Laplace Transforms to solve differential equation initial value problems is a great way to streamline solutions and, for forcing functions that are discontinuous, they are about the only way to solve them. The equations we use are
Laplace Transforms of Derivatives 

\(\displaystyle{ \mathcal{L}\{y'\} = sY(s)  y(0) }\) 
\(\displaystyle{ \mathcal{L}\{y''\} = s^2Y(s)  sy(0)  y'(0) }\) 
where \(\mathcal{L}\{y(t)\} = Y(s)\) 
\(\displaystyle{ \mathcal{L}\{ f^{(n)}(t) \} = }\) \(\displaystyle{ s^nF(s)  s^{n1}f(0)  }\) \(\displaystyle{ s^{n2}f'(0)  . . .  f^{(n1)}(0) }\) 
The idea is that we are given a differential equation which we first transform using the Laplace Transform, we solve it in the sdomain and then perform an inverse Laplace transform back to the tdomain to get our answer. Here is a very quick video giving these steps in a flowchart.
PatrickJMT  Laplace Transform flowchart  
This video clip explains in detail how to solve differential equations using Laplace Transforms including a quick calculation of the Laplace transform of the first derivative function above.
Dr Chris Tisdell  solve differential equations using Laplace Transforms  
This video expands on the one above taking you stepbystep through this process and, at the end of the video, gives you a big picture of how to do this. This is a great video.
Dr Chris Tisdell  big picture  
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Practice Problems 

Instructions   Unless otherwise instructed, solve these initial value problems using Laplace transforms. Give your answers in exact, completely factored form.
In the following problems, \(u(t)\) is the unit step function (Heaviside function).