Simple Differential Equation: Separation of Variables. Please help!?

Question

I'm trying to solve this differential equation. Any help would be appreciated. Thank you!

(x+x*(y^2))dx + e^(x^2)*ydy = 0

I separated the variables and integrated, but my solution doesn't look right. Thanks in advance!

Answer 1

It would be better if you posted what you think the solution is, then people can help point out where you went wrong (if, indeed, you did).

As you say, this is a separable equation:

(x+x*y^2) dx = -y*exp(x^2) dy

x*(1+y^2) dx = -y*exp(x^2) dy

x*exp(-x^2) dx = [-y/(1+y^2)] dy

(-1/2)*exp(-x^2) = (-1/2)*ln(1+y^2) + C

define the constant of integration, C = -ln(a), where a is some other constant, and cancel the factor of -1/2 on each side of the equation to get:

exp(-x^2) = ln((1+y^2)/a)

exp(exp(-x^2)) = (1+y^2)/a

y^2 = a*exp(exp(-x^2)) - 1

y = +sqrt(a*exp(exp(-x^2)) - 1) and y = -sqrt(a*exp(exp(-x^2)) - 1)