Almost Separable Differential Equation

[MDR123]

(Moderator's note: thread moved from "Differential Equations")

Hi, I am trying to solve a differential equation which is almost separable, but not quite.

Specifically:

[tex]1 + \frac{dy}{dx} = f(x) g(y)[/tex]

Is there any way to approach this, or are additional constraints needed? In the setting in which I am working, constraints on f would be much more natural than constraints on g.

Thank you very much,
Michael

PS. I could make a substitution to replace y with y - x to get rid of the 1 on the LHS, but then dealing with the RHS is not so clear to me. Alternately, I've tried differentiating both sides so as to get rid of the 1, but that leaves me with a second order problem. Alternately, I could divide both sides by and then differentiating leaves me with only y' , y'' terms, but alas, it is again nonseparable.

 

[lippyka]

One approach you could take is to use the integrating factor method. The idea is to multiply both sides of the equation by an integrating factor, M(x), and then find an expression for M(x) such that the resulting equation is separable.If we let M(x) = exp[∫f(x)dx], then the equation can be written as M(x) + M(x)y' = f(x)g(y).Now, if we divide both sides by M(x), we obtain1 + y' = \frac{f(x)g(y)}{M(x)}which is a separable equation. Therefore, we can solve it by integrating both sides with respect to x.