# How does one solve the differential equation $y'=\frac {y} {3x-y}$ ?

I need to find a solution to the differential following equation:

$y'=\frac {y} {3x-y}$.

I tried to use use some kind of substitution, but I didn't manage to solve it.

Any suggestion\help?

Thanks a lot!

• You lost me here $y'=\frac {y} {3x-y} \Rightarrow 3x-y^2=y$. Maybe this is using techniques I'm not familiar with, but this seems wrong. Mar 23, 2012 at 21:54
• Sorry, my mistake. Mar 23, 2012 at 22:04
• Perhaps this. Mar 23, 2012 at 22:05
• It seams that $y=ax$ could be a solution. Mar 23, 2012 at 22:07
• See $y'$ is essentially ${dy \over dx}$. So just do ${dx \over dy}$ and then try to solve it. It'll be finished in a matter of seconds! Mar 15, 2022 at 7:40

Hint: Try $y=zx$. That will let you separate variables.

• You mean the denominator is $3x-y^2$? I don't know, nonlinear equations are difficult. One might be able to come up with a special trick reducing to a Riccati, but a little scribbling got me nowhere. In the original problem, taking advantage of homogeneity was an automatic reflex. Mar 24, 2012 at 16:06

We can transform this equation to linear differential equation.

$\frac{dx}{dy}= \frac{3x}{y}-1$

And integrating factor $\lambda=e^{{\int{\frac{-3}{y}dy}}}=y^{-3}$.

Then $x=\frac{\int(-1)y^{-3}dy}{y^{-3}}$. Therefore $x=cy^{3}+\frac{y}{2}$.

• If you care to, you can explicitly solve for $y$ in terms of $x$ when $c\geq0$. If $c=0$, then of course $y=2x$. If $c>0$, then the cubic formula yields $y=\sqrt{ \frac{x}{2c}+\sqrt{\frac{x^2}{4c^2}+\frac{1}{216c^3}}}+\sqrt{ \frac{x}{2c}-\sqrt{\frac{x^2}{4c^2}+\frac{1}{216c^3}}}$. Mar 24, 2012 at 0:03

I followed $y=z.x$ transform and I got easily the general solution : $cy^3+y=2x$

$3cy^2y'+y'=2$

$(3cy^3+y)y'=2y$

$(3.(2x-y)+y)y'=2y$

$(6x-2y)y'=2y$

$(3x-y)y'=y$

$y'=\frac{y}{3x-y}$

If you cannot manage the y=z.x transform ,let me know.

$$y' = \frac{y}{3x - y} = \frac{1}{\frac{3x}{y} - 1}$$

$$\frac{y}{x} = v \Rightarrow y = vx \Rightarrow \frac{dy}{dx} = x \frac{dv}{dx} + v$$

$$x \frac{dv}{dx} + v = \frac{1}{\frac{3}{v} - 1} = \frac{v}{3 - v}$$

$$x \frac{dv}{dx} = \frac{v}{3-v} - v = \frac{v(v-2)}{3-v}$$

$$\frac{(3-v)dv}{v(v-2)} = \frac{dx}{x}$$

$$\Rightarrow \int \frac{(3-v)}{v(v-2)} \ dv = \int \frac{dx}{x} = \ln x + c$$

$$\frac{(3-v)}{v(v-2)} = \frac{A}{v} + \frac{B}{v-2}$$

$$\Rightarrow A(v-2) + Bv = 3 - v$$

$$v = 2 \Rightarrow B = \frac{1}{2}$$

$$v = 0 \Rightarrow A = -\frac{3}{2}$$

$$\Rightarrow \int \frac{(3-v)}{v(v-2)} \ dv = -\frac{3}{2} \int \frac{1}{v} \ dv + \frac{1}{2} \int \frac{1}{v-2} \ dv = \frac{1}{2} \ln(v-2) - \frac{3}{2} \ln v = \ln x + c$$

$$v = \frac{y}{x} \Rightarrow \frac{1}{2} \ln \left(\frac{y}{x}-2 \right) - \frac{3}{2} (\ln y - \ln x) = \ln x + c$$