Solving a Differential Equation with Integrating Factors

[CalculusHelp1]

Homework Statement

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

Homework Equations

definition 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

 

[fzero]

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 $$y=y_h + y_p$$.

 

[SammyS]

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"