solve non linear differential equation: $y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0$

Question

Could somebody help me to solve the non linear differential equation, where $y$ is a function of the time and starts with $y(0)=0$ $$ y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0 $$ It will guess it will involve the Lambert W functions.

Answer 1

your equation is of the form

$ y'= f(y) $ this can be expressed in the form (implicit)

$$ x= \frac{1}{a}\int \frac{dy}{-d-be^{cy}-y} $$

of course hre $ a= \alpha $ $b= \beta $ and $c= \delta $ and $ d= \theta $to simplify terms

i do not know how to integrate the expression i guess the integral can not be obtained exactly