# Solve Linear ODE Using Integrating Factor

1. Sep 12, 2013

### 1s1

1. The problem statement, all variables and given/known data

Solve the initial value problem:
$$sin(x)y' + ycos(x) = xsin(x), y(2)= \pi/2$$

2. Relevant equations

3. The attempt at a solution

Recognizing it as a Linear First-Order Equation:$$\frac{dy}{dx}+y\frac{cosx}{sinx}=x$$
$$P(x)=\frac{cosx}{sinx}$$
Integrating factor: $$e^{\int \frac{cosx}{sinx}dx}=sinx$$

Multiplying the ODE by the integrating factor:
$$\frac{d}{dx}[ysinx] = xsinx$$

Integrating both sides: $$ysinx = \int xsinx dx$$
$$y=1-\frac{xcosx}{sinx}+C$$
Solving for C: $$C=1$$
$$y=2-\frac{xcosx}{sinx}$$

Apparently this solution is incorrect, but I can't figure out why?

2. Sep 12, 2013

### rock.freak667

From this line

ysinx=∫xsinxdx

you should get

ysinx = sinx - xcosx +C

then divide by sinx.

3. Sep 12, 2013

### 1s1

Thanks! When I divided through by $sin(x)$ from the step you suggested, I had forgotten to divide $C$ by $sin(x)$. I tend to ignore the constants which is a big mistake. Thanks for the help!

4. Sep 12, 2013

### epenguin

maybe you have missed out some steps of what you did, it is not evident that you have worked out your integration factor or multiplied by it.

You don't need an integration factor and can go straight from line 1 to line 5.