# Differential Equation

CalculusHelp1

## Homework Statement

Find the solution to (x+1)y' +2y -(x+1)^(5/2)=0

## Homework Equations

Definiton of antiderivative

## The Attempt at a Solution

I have been trying to manipulate this equation in every possible way so that I can get x on one side and y on the other. Ever attempt has led to a dead end. I tried to factor the (x+1) terms, tried to carry things over to the other side, nothing is working.

Can anyone give me a nod in the right direction so I can tackle this problem? Thanks

Homework Helper
Gold Member
For a particular solution, you might try

$$y_p = a (x+1)^b,$$

where there's an obvious guess for $$b$$ if you think a bit before doing any calculations. Otherwise, plug this in and see what values of $$b$$ and $$a$$ solve the equation.

To get the full solution, you also have to solve the homogeneous equation

$$(x+1)y' +2y =0$$

which will give you a function $$y_h(x)$$. The general solution to the original equation is $$y=y_h + y_p$$.

Staff Emeritus
Homework Helper
Gold Member
Multiply by (x+1), then notice that:

$$\frac{d}{dx}\left((x+1)^2y\right)=(x+1)^2y'+2(x+1)y$$

CalculusHelp1
Okay this problem might be over my head. I've only just learned first order linear differential equations and solving them by separation of variables.

Is there any easier way to do this?

Robert1986
Okay this problem might be over my head. I've only just learned first order linear differential equations and solving them by separation of variables.

Is there any easier way to do this?

Have you learned about integrating factors?

CalculusHelp1
No, what are those?

Robert1986
No, what are those?

I could give a reasonably good answer, but Paul of Paul's Online Math Notes does a better job. Just Google : "integrating factors, pauls online math notes"