# Need help solving this differential equation

I've separated the variables of this differential equation and end up with
dx/((a-x)^(1/2)*(b-c(x-d)^3/2)). I've tried finding the integral of this with non-trig substitution methods but cannot solve it. Any help would be appreciated.

eumyang
Homework Helper
I've separated the variables of this differential equation and end up with
dx/((a-x)^(1/2)*(b-c(x-d)^3/2)).
I don't see an equation here. (Where's the equals sign?) Start by writing the original problem and show us what work you have.

The original diff eq is

dx/dy = b(a-x)^1/2 - c(a-x)^1/2 * (x-d)^3/2

Separating variables results in my original posted equation

dy = dx/((a-x)^1/2 * (b-c(x-d)^3/2))

I have tried the substitution, u = (a-x)^1/2, x = a-u^2, dx = -2udu. Which results in

dy = -2dx/(b-c((-u^2+a)-d)^3/2)

Any further non-trig substitutions does not help to simplify. I believe a trig substitution is required but I have little experience with trig subs. I have already put any many hours looking for a solution and would like to know if a trig sub could be used to solve this. Thanks

haruspex
Homework Helper
Gold Member
2020 Award
I don't think there's any point in manipulating the ODE. You have reduced it to an integral, so work with that. Your substitutions so far look good, but I believe you can simplify it to ##\frac{du}{A-(1-u^2)^{\frac32}}##. Substituting u = sin(θ) and expanding with partial fractions can get you to a sum of terms like ##\frac{d\theta}{W-cos(\theta)}##, but I don't know where to go from there.

I'm not seeing anywhere to go after that either. But thanks for the input.

Ray Vickson