# general solution to non linear ordinary differential equation

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.

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

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 $$-\frac{3 }{4 \sqrt{7}}\tan ^{-1}\left(\frac{2 v+1}{\sqrt{7}}\right)+\frac 14\log \left(\frac{(v-1)^2}{v^2+v+2}\right)$$