Homogeneous polar equation substitutions

Hello all,

I've spent the better part of an hour on a single problem. I've attempted it several ways but have yet to find the right answer - every attempt is resulting in a different answer.

The method I'm supposed to be using is substitution. The problem is (2x^2-3xy)y'=x^2+2xy-3y^2

The correct answer is y(x)=x*|ln(y^2/x)+C|.

My latest answer is y=2*x*ln(y/x) - x*ln|x| as a result of integrating (2-v)dv = -(1/x)dx where v=y/x. My previous two attempts had vastly different answers. I have a sneaking suspicion my problem is in my algebra before I start integrating.

Re: Homogeneous polar equation substitutions

We are given to solve:

$\displaystyle \left(2x^2-3xy \right)y'=x^2+2xy-3y^2$

$\displaystyle \frac{dy}{dx}=\frac{x^2+2xy-3y^2}{2x^2-3xy}=\frac{1+2\left(\frac{y}{x} \right)-3\left(\frac{y}{x} \right)^2}{2-3\left(\frac{y}{x} \right)}$

Now, if we use the substitution:

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

We may write:

$\displaystyle v+x\frac{dv}{dx}=\frac{1+2v-3v^2}{2-3v}$

$\displaystyle x\frac{dv}{dx}=\frac{1+2v-3v^2}{2-3v}-v=\frac{1+2v-3v^2-v(2-3v)}{2-3v}=\frac{1}{2-3v}$

$\displaystyle \int 2-3v\,dv=\int\frac{1}{x}\,dx$

$\displaystyle 2v-\frac{3}{2}v^2=\ln|x|+C$

Now, back-substitute for $\displaystyle v$, however, I don't see how the given result is correct.

Re: Homogeneous polar equation substitutions

I miss-scribed from my book - that's actually the problem right next to the problem I'm having the problem with! (2x^2-xy)y'=xy-y^2 is the problematic problem.

I believe my problem to be in the algebra - I'm rusty from not doing anything algebraically tricky for a few years.

In the problem I gave you, your second line, to the right of the second equal sign, you smack the problem upside the head with some algebra - you multiply through with 1/x^2, correct?

In my problem it appears I should do the following (Pardon my lack of Latex):

y'=(x^3+3y^3)/(2x^2-xy) * (1/x^3)/(1/x^3)

= (1+3(y^3/x^3))/((2/x)-((y/x)*(1/x))

v=y/x

y=vx

dy/dx=v+x(dv/dx)

v+x(dv/dx)=(1+3v^2)/((2/x)-(v/x) = (x^2*(3v^3+1))/(2x-v)

At about this point, I think I'm lost again. I expanded what I had and spent half a page algebraically manipulating it.

This is what I boiled it down to, likely incorrectly: 2/x-v/(x^2) + (dv/dx)*(2/v - 1) = 3v^2+1/v.

Re: Homogeneous polar equation substitutions

The first thing I notice about $\displaystyle (2x^2- xy)y'= xy- y^2$ is that all terms are of second degree- $\displaystyle x^2$, $\displaystyle y^2$, or $\displaystyle xy$. That suggests the substitution $\displaystyle u= \frac{y}{x}$ or $\displaystyle y= xu$. Then $\displaystyle y'= xu'+ u$ and the equation becomes $\displaystyle (2x^2- x^2u)(xu'+ u)= x^2u+ x^2u^2$. Dividing through by $\displaystyle x^2$ $\displaystyle (2- u)(xu'+ u)= (-xu+ 2x)u'+ 2u- u^2= u- u^2$ or $\displaystyle x(2- u)u'= -u$, which is separable: $\displaystyle \frac{(2- u)du}{-u}= (1- \frac{2}{u})du= \frac{dx}{x}$.