Question

How do you find all solutions of the differential equation `dy/dx=xe^y`?

Answer 1

`y = ln (1/(A-1/2x^2))`

Explanation:

The equation

`dy/dx = xe^y`

is a First Order linear separable Differential Equation which can be solved simply by rearranging and collection term in `x` on the RJS and term in `y` on the LHS;

`1/e^ydy/dx = x => e^-ydy/dx = x`

And now we "separate the variables" to get;

`int \ e^-y \ dy = int \ x \ dx`

Which is trivial to integrate to get:

`-e^-y = 1/2x^2 - A \ \ \ \` (I've chosen `c=-A` as the integration constant)
`:. e^-y = A-1/2x^2`
`:. -y = ln(A-1/2x^2)`
`:. y = -ln(A-1/2x^2)`
`:. y = ln (1/(A-1/2x^2))`