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Hi all I am building a computer program which needs a general solution to the following differential equation.

$$1 - \frac{dz}{dy} = (z/y + 1)^r$$

In this equation $r$ is a positive constant, $z > 0$, and $y > 0$.

I have tried solving this as a homogenous differential equation where I set $u = z/y$ doing this results in the following equation $$y = C * e^{-\int_1^{\frac{z}{y}}\limits{\frac{du}{(u+1)^r + u - 1}}}$$

I know that I can approximate a result though due to computational limits of my computer program this is impractical and a general solution would be much better. I have tried feeding this equation to python sympy, wolfram mathematica, as well as matlab and none of them were able to find a solution. If anyone has any insight into this problem I would really appreciate hearing your feedback.

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  • $\begingroup$ This is a homogeneous equation. Call $u=z/y$ and solve for $u(x)$. $\endgroup$
    – GReyes
    Dec 15, 2020 at 19:32

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Starting from @Joshua Wang's answer (now deleted) $$\frac{dz}{z} = \frac{du}{1 - u - (u + 1)^{r}}$$ let $u=v-1$ to make $$\frac{dz}{z} =- \frac{dv}{v^r+v-2}$$ If $r$ is an integer (otherwise, I do not think about a possible solution), we can write $$\frac{dz}{z} =- \frac{dv}{\prod_{k=1}^r(v-s_k)}$$ where the $s_k$'s are the roots of the polynomial $v^r+v-2=0$.

Partial fraction decomposition $$\frac{dz}{z} =- \sum_{k=1}^r\frac{a_k}{v-s_k}\,dv$$

$$\log(|z|)+C=- \sum_{k=1}^r a_k \log(|v-s_k|)$$ and we shall probably end with a sum of logarithms and arctangents.

For $r=3$, the rhs would be $$-\frac{3 }{4 \sqrt{7}}\tan ^{-1}\left(\frac{2 v+1}{\sqrt{7}}\right)+\frac 14\log \left(\frac{(v-1)^2}{v^2+v+2}\right)$$

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  • $\begingroup$ unfortunately I need to deal with the possibility that r may be a decimal $\endgroup$ Dec 17, 2020 at 15:25

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