# Finding the general solution of the D.E.

• Jun 2nd 2011, 05:38 PM
bambamm
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.
• Jun 2nd 2011, 05:41 PM
Ackbeet
Try this:

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

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

Can you see what to do from here?
• Jun 3rd 2011, 03:19 AM
HallsofIvy
But I do like your idea of treating x and the dependent variable.
$\displaystyle \frac{dx}{dy}= x+ ye^y$
is a fairly straight forward linear equation.
• Jun 3rd 2011, 12:53 PM
bugatti79
Quote:

Originally Posted by HallsofIvy
But I do like your idea of treating x and the dependent variable.
$\displaystyle \frac{dx}{dy}= x+ ye^y$
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
• Jun 3rd 2011, 12:58 PM
Ackbeet
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.