# Thread: Finding the general solution of the D.E.

1. ## Finding the general solution of the D.E.

Q. Find the general solution of (x+y*e^y)*dy/dx = 1.

I'm having trouble solving this. I know you have to make this D.E. linear by interchanging dx and dy. So you have to solve for x(y) and treat y as the independent variable and x as the dependent variable, but I don't know how to proceed from here. Any guidance would be greatly appreciated.

2. Try this:

$(x+y e^y)\frac{dy}{dx} = 1$

$\frac{dy}{dx}=\frac{1}{x+ye^{y}}$

Can you see what to do from here?

3. But I do like your idea of treating x and the dependent variable.
$\frac{dx}{dy}= x+ ye^y$
is a fairly straight forward linear equation.

4. Originally Posted by HallsofIvy
But I do like your idea of treating x and the dependent variable.
$\frac{dx}{dy}= x+ ye^y$
is a fairly straight forward linear equation.
May I ask is this done via a substitution? I am looking at a math book here which say the method of substitution can be use for 'homogeneous equations', but this is not homogeneous, right?

Thanks

5. No, it's not homogeneous. But it is linear in x, and hence you can use the standard integrating factor method, as HallsofIvy hinted at.