Coupled non-linear ODE

Question

I have a sub-ode in a larger system that I am working with. It takes the form of

$$y' = a-b\frac{y}{x}$$ $$x'=c\frac{y}{x}-d$$

I've been looking around for solutions since I do not know of a general method for non-linear coupled ODEs. Does anyone have some intuition about the above system. It seems like there must exist a nice solution since $\frac{y}{x}$ appears in both equations.

Thanks!

Edit: This is under a known constraint that $x+y=C$

Edit 2: In the actual problem $b=c$.

Answer 1

Hint: notice that

$$y'+\frac{b}{c}x' = a-\frac{bd}{c} \implies bx+cy = (ac-bd)t+k$$

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$\textbf{EDIT}$: With the edits we can simplify the equation to

$$x' = \frac{bC-(a+b)x}{x}$$

which is separable. It has a solution of

$$\frac{bC}{(a+b)^2}\log(bC-(a+b)x) + \frac{x}{a+b} = K-t$$

where $K$ is the constant of integration.

Answer 2

Hint.

$$ (x+y)'=a-b\frac yx-c\frac yx-d = (a-d)+(b+c)\frac yx = 0\Rightarrow a-d = 0,\ \ \ b+c=0 $$

the the system to solve is

$$ \cases{x' = a - b\frac yx\\ y'= b\frac yx - a} $$

NOTE

MATHEMATICA gives the result

$$ \cases{ x(t) = \frac{b c_1}{a+b}\left(W\left(\frac{e^{\left(\frac{c_2 (a+b)^2+t (a+b)^2-b c_1}{b c_1}\right)}}{b c_1}\right)+1\right)\\ y(t) = c_1-\frac{b c_1}{a+b}\left(W\left(\frac{e^{\left(\frac{c_2 (a+b)^2+t (a+b)^2-b c_1}{b c_1}\right)}}{b c_1}\right)+1\right) } $$

Here $W(\cdot)$ is the Lambert function.