# Thread: find the integrating factor and then use it to derive the general solution y(x)

1. ## find the integrating factor and then use it to derive the general solution y(x)

Find the integrating factor of the equation: [e^y*x^2*cosx+1]dx+[e^y*x^2*sinx]dy=0 and use the integrating factor to find the general solution. Please help!

2. Hint : The equation has an integrating factor that depends on x .

3. So if I let M=e^y*x^2*cosx+1, then M with respect to y is x^2*e^y*cosx+y
If N=e^y*x^2*sinx, then N with respect to x is e^y(2xsinx-x^2+2cosx) right?

So if I use (Ms of y-Ns of x)/N=((x^2*e^y*cosx+y)-(e^y(2xsinx-x^2+2cosx))/(e^y*x^2*sinx).

But it's so hard to simplify this to get the integrating factor. Or am I doing something wrong?

4. We have

( 1 / N ) ( M_y - N_x ) = ... = - 2 / x

so, the integrating factor m ( x ) satisfies

m ' ( x ) / m ( x ) = - 2 / x

5. wait, so you got -2/x after dividing (M_y-N_x) by N?

6. Originally Posted by Taurus3
wait, so you got -2/x after dividing (M_y-N_x) by N?

And you should get the same.