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

The integral is VERY complicated. It involves a lengthy formula that uses the roots of a 6th degree polynomial whose coefficients are functions of a, b, c and d. I did the integral in Maple, but if you do not have access to Maple you could try to submit it to Mathematica. Wolfram Alpha failed to find the integral.

Glad to hear that there is a solution. Since this thread is no longer in the schoolwork forum, would you be more forthcoming with the solution.