# 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?

##### 2 Answers

#### Explanation:

The differential equation is first order linear nonhomogeneus. In this case the solution is composed from the homogeneus solution

Finally

1) Obtaining the homogeneus solution

Simplifying we obtain

grouping variables

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

https://en.wikipedia.org/wiki/Variation_of_parameters

Introducing

Solving for

Now, putting all together

With the initial conditions we find the

#### Explanation:

this is linear so we can start by just moving stuff around.

we solve with inregrating factor

From