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solve for $x,y,z$: $\frac{dx}{x^{2}+a^{2}}=\frac{dy}{xy-az}=\frac{dz}{xz+ay}$

please give a hint. I am not able to formulate the steps required to proceed solving this one.

2 Answers 2

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We have $ \begin{align} \frac{dy}{xy-az} & = \frac{dz}{xz+ay}\\ \frac{dy/y}{x-a (z/y)} & = \frac{dz/z}{x+a (y/z)} \end{align} $ This gives a motivation to let $z = ky$ where $k$ is a constant. $ \begin{align} \frac{dy/y}{x-a k} & = \frac{dy/y}{x+a/k} \end{align} $ This gives us that $k = \pm i$. Let $k=i$. This gives us that $\begin{align} \frac{dx}{x^2 + a^2} & = \frac{dy/y}{x - ia}\\ \frac{dx}{x + ia} & = \frac{dy}{y}\\ y & = c(x + ia) \end{align} $ Hence, we get $ \begin{align} z & = ic(x+ia)\\ y & = c(x+ia) \end{align} $ and $ \begin{align} z & = -ic(x-ia)\\ y & = c(x-ia) \end{align} $ I don't know to justify my motivation why I chose $z = ky$ instead of $z=k(y)y$.

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Since the request is for a hint, I promote my comment to an answer. Write your system as $\frac{dy}{dx}=\frac{xy-az}{x^{2}+a^{2}}$

$\frac{dz}{dx}=\frac{xz+ay}{x^{2}+a^{2}}$ Maple does show non-constant solutions for this.