4.8.1 Introduction

This note is about solving a first order ode of the form $y^{\prime }={(a+bx+cy)}^{\frac{1}{n}}$ and $y^{\prime }={(a+bx+cy)}^m$ where $n,m\ne 1$ and are integers. This is of the form  $y^{\prime }=f{(x,y)}^{\frac{1}{n}}$ and $y^{\prime }=f{(x,y)}^m$. Where $f(x,y)$ must be linear in both $y$ and $x$. The reason it needs to be linear in $x$ so that the transformed ode in $z$ becomes separable.

One way to solve $y^{\prime }={(a+bx+cy)}^{\frac{1}{n}}$ is to raise both sides to $n$. For example for $n=2$ the ode becomes ${\left(y^{\prime }\right)}^2=(a+bx+cy)$ which can be solved as d’Alembert.

This is what Maple seems to do based on what the Maple advisor says about the type of this ode being d’Alembert.

But the problem with squaring both sides or raising both sides of ode to some power is that this will introduce extraneous solutions to the original ode. Hence it is will be better to avoid doing this if at all possible.

The following methods solve these odes without having to square or raise both sides to same power and eliminate the introduction of extraneous solutions.

It is important to note that $f(x,y)$ must be linear in $x,y$ and not have product terms $xy$.