Is Your Non-linear DE Still Not Exact After Integrating Factor Adjustment?

[ifly2hi]

Cannot prove the DE is exact??

Hey guys I am looking at a non-linear first order DE. the problem is : (y^2)/2+2*y*exp^(x) +(y+exp^(x))dy/dx=exp^(-x) y(0)=1. I put everything on the same side: ((y^2)/2+2*y*exp^(x)-exp^(-x))dx+(y+exp^(x))dy=0. This equation is not exact so I use (My-Nx)/N and got a function of x alone that equaled 1. Put it into exp^(∫1dx)=exp^(x); took this and multiplied it my N and still M=(y^2)/2+2*y*exp^(x)-exp^(-x) and the "new" N=exp^(2*x)+y*exp^(x): and still My≠Nx. It is still not exact and i have no idea where to go from here? help

 

[vela]

Please use the template when posting here, and try to make your post easier to read. I did a little reformatting and clean-up for you.

Homework Statement

Hey guys. I'm looking at a non-linear, first-order DE. The problem is:
$$\frac{y^2}{2} + 2ye^x + (y+e^x)\frac{dy}{dx} = e^{-x}$$ with y(0)=1.

Homework Equations

(M_y-N_x)/N

The Attempt at a Solution

I put everything on the same side:
$$\left(\frac{y^2}{2} + 2ye^x-e^{-x}\right)dx + (y+e^x)dy = 0.$$This equation is not exact, so I used (My-Nx)/N and got a function of x alone that equaled 1. Put it into exp^(∫1dx)=exp^(x); took this and multiplied it my N and still M=(y^2)/2+2*y*exp^(x)-exp^(-x) and the "new" N=exp^(2*x)+y*exp^(x). Still M_y≠N_x. It is still not exact and i have no idea where to go from here? help

You don't multiply N and not M by the integrating factor. You have to multiply both because you're multiplying the equation by the integrating factor.
$$e^x\left(\frac{y^2}{2} + 2ye^x-e^{-x}\right)\,dx + e^x(y+e^x)\,dy = 0$$ If you check the new M and N in that equation, you'll find it's now exact.

 

[ifly2hi]

Yeap... that was a rookie mistake on my part. I do have another question: I have solved the problem down to the point where ∂F/∂x=(2ye^2x+y²e^x-2x)/2+A(y) and ∂F/∂y=e^x(y+e^x)→e^2x+A'(y) am I correct in answering A'(y)=ye^x because it is the only variable of y in ∂F/∂y which i would then intergrate to find A(y)=(y²e^x)/2 + C and put this back in the equation for ∂F/∂x and solve to the constant C?

 

[vela]

Can you show your work in more detail? I'm not sure what you're doing. Your expression for $\partial F/\partial x$ isn't correct, and I'm not sure where you got $e^{2x}+A'(y)$ from.