What is a solution to the differential equation #(x+1)y'-2(x^2+x)y=e^(x^2)/(x+1)# where x>-1 and y(0)=5?
The differential equation is first order linear nonhomogeneus. In this case the solution is composed from the homogeneus solution
1) Obtaining the homogeneus solution
Simplifying we obtain
The solution is
2) Obtaining the particular solution
For this purpose we will suppose that
This method was due to Euler and Lagrange and can be seen in
Now, putting all together
With the initial conditions we find the
this is linear so we can start by just moving stuff around.
we solve with inregrating factor