Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

2 Answers 2

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 $z=k(y)y$.

share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.