Solving a differential equation with intial conditions,

[skyturnred]

Homework Statement

Here is the original thing:

(x^{2}+1)y'+4x(y-1)=0, y(0)=4

Homework Equations

The Attempt at a Solution

I thought I knew the procedure.. but I got it wrong. Can someone let me know where I went wrong?

First I rearrange the equation to get the following in the form y'+p(x)y=g(x)

y'+(\frac{4x}{x^{2}+1})y=\frac{4x}{x^{2}+1}

So I get the integrating factor to be

I(x)=e^{\int\frac{4x}{x^{2}+1}}

That integral comes out to be 2ln(x^{2}+1), which when raised to e, the integrating factor simply becomes

I(x)=(x^{2}+1)^{2}

So the new differential equation is:

d/dx (x^{2}+1)^{2}y=\frac{4x}{x^{2}+1}

integrating both sides gets me

(x^{2}+1)^{2}y=2ln(x^{2}+1)+C

solving for y:

y=\frac{2ln(x^{2}+1)}{(x^{2}+1)^{2}}+\frac{C}{(x^{2}+1)^{2}}

Taking the intial conditions into account I get C=4. When I input this equation into my assignment online, it says I'm wrong.

Please help, I thought I knew this process pretty well..

 

[vela]

You forgot to multiply the righthand side by the integrating factor.

 

[HallsofIvy]

Homework Statement

Here is the original thing:

(x^{2}+1)y'+4x(y-1)=0, y(0)=4

Homework Equations

The Attempt at a Solution

I thought I knew the procedure.. but I got it wrong. Can someone let me know where I went wrong?

First I rearrange the equation to get the following in the form y'+p(x)y=g(x)

y'+(\frac{4x}{x^{2}+1})y=\frac{4x}{x^{2}+1}

So I get the integrating factor to be

I(x)=e^{\int\frac{4x}{x^{2}+1}}

That integral comes out to be 2ln(x^{2}+1), which when raised to e, the integrating factor simply becomes

I(x)=(x^{2}+1)^{2}

So you are going to multiply both sides of the equation by that?

So the new differential equation is:

d/dx (x^{2}+1)^{2}y=\frac{4x}{x^{2}+1}

Did you not multiply the right side by (x^2+ 1)^2?

integrating both sides gets me

(x^{2}+1)^{2}y=2ln(x^{2}+1)+C

solving for y:

y=\frac{2ln(x^{2}+1)}{(x^{2}+1)^{2}}+\frac{C}{(x^{2}+1)^{2}}

Taking the intial conditions into account I get C=4. When I input this equation into my assignment online, it says I'm wrong.

Please help, I thought I knew this process pretty well..