Differential equation with two dependent variables

Question

I have following differential equation -

$z'yx + y'zx + yz - ay'=0$

where, $y$ and $z$ both depend on $x$.

$y'= \frac{dy}{dx}$ and $z'= \frac{dz}{dx}$

$a$ is constant

I have to find $y(x)$ and $z(x)$. I have just this equation to work around a solution. Can I solve it? How to solve this differential equation with two dependent variables?

Any topic suggestion or hint will be helpful

Answer 1

Hint: You can rewrite the differential equation like so

$$\dfrac{d}{dx}\left[xy(x)z(x) \right] = \dfrac{d}{dx}\left[ay(x)\right].$$

Integration will yield

$$xy(x)z(x) = ay(x) + c$$ $$\implies \left[xz(x)-a\right]y(x) = c.$$

Edit after additional information was provided: If we know that $z(0)=y(0)=0$ then we can conclude

$$\left[0 \cdot z(0) - a\right]y(0) = c.$$ $$0 = c$$

This will imply

$$\left[xz(x)-a\right]y(x) = 0.$$

A product can only be zero if at least one of the terms is zero.

In total we can have the following conditions

• Only first term is $0$: $xz(x)-a = 0 \implies z(x)=a/x$ and $y(x)$ is arbitrary.
• Only second term is $0$: $y(x) = 0$ and $xz(x) - a$ is arbitrary.
• Both terms are $0$: $y(x) = 0$ and $z(x)=a/x$ (is included in the first case).