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Problem 1: Use variation of parameters to find the general solution to
First we need to solve the homogeneous equation
y''' - y' = 0. Its characteristic equation is 
. It is easy to see that its root
are r=0,1,-1. Therefore we have 
.
Second we need to find a particular solution using the variation of parameters technique.
We have 
, where u', v', and w' are solution to the system
Easy calculations give
Integration by parts, will give (you are encouraged to do it)
Hence we have
which implies that 
once u, v, w are plugged into the formula
giving 
. Therefore the general solution is given by:
Problem 2. Find the solution to the initial value problem
where
Since g(t) is a step function, we need to use Laplace Transform to solve this problem. Once we attack the equation by 
, we get
which implies 
. Using the initial condition, we get
After easy calculations, we obtain
Since 
, we deduce
Hence
In order to find y, we need to use the inverse Laplace transform. First we have
which implies
The hard part concerns the second term 
. First let us find the inverse Laplace without the exponential. First we know that
Using the derivative formula, we get
Therefore, we have
Using the formula 
, we get
Therefore, we have
Problem 3. Find the Laplace transform of
We have
Hence 
. Using the formula
, we get
But 
which implies
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