Non-linear first order ODE: Solving with Exact Form

[estro]

I'm trying to solve the following ODE: $$ydx+(\frac {e^x}{y}-1)dy=0$$

I tried to transfer this ODE into exact form but no luck.

Will appreciate any help.

 

[LCKurtz]

I'm trying to solve the following ODE: $$ydx+(\frac {e^x}{y}-1)dy=0$$

I tried to transfer this ODE into exact form but no luck.

Will appreciate any help.

Leave the equation as it is and look for an integrating factor that is just a function of x.

 

[estro]

I tried it before:
$$X=y, Y=e^x/y-1$$
$$u'(x)+(\frac {X_y-Y_x}{Y})u(x)=0$$
Thus u=e^x

But $$d/dy(e^xX) \neq d/dx(e^xY)$$

What I'm missing?

 

[LCKurtz]

I tried it before:
$$X=y, Y=e^x/y-1$$
$$u'(x)+(\frac {X_y-Y_x}{Y})u(x)=0$$

Are you sure about the sign on that numerator?

 

[estro]

Bellow is formula that I found in my book:
$$u'(x)+(\frac {X_y-Y_x}{Y})u(x)=0$$=
$$u'(x)-u(x)=0$$ => u=e^x

Can't see where is the problem.

 

[LCKurtz]

Are you sure about the sign on that numerator?

Bellow is formula that I found in my book:
$$u'(x)+(\frac {X_y-Y_x}{Y})u(x)=0$$=
$$u'(x)-u(x)=0$$ => u=e^x

Can't see where is the problem.

I told you what was wrong. The numerator should be Y_x-X_y.