This is the question:
Solve $$x^2 \frac{\text{d}y}{\text{d}x} = y^2 - 3xy -5x^2 .$$ Are there any straight line solutions? If so, state them.
For the case $y(1)=0$, state the solution explicitly, and describe (with appropriate justification) what the solution looks like:
when $|x|$ is very large;
when $x$ is very close to zero.
I tried to solve the first part:
Did i solve it right? Assuming i did, how do i know if there are any straight line solutions? And how do i got about doing the other parts?
Should i set $c=0$ to find if there are straight line solutions? I know there is a way using $y=mx$, so $dy/dx = m$, but i'm not sure please help
Thanks
Your calculations look OK, but seem to miss the point. You do not need to compute the full solution, you only need to test solutions of the required format.
The equation is homogeneous of degree 2 in the polynomial sense. Thus set $y=xu$ then $$ x^2y'=x^3u'+x^2u=x^2(u^2-3u-5)\implies xu'=u^2-4u-5=(u-5)(u+1) $$ This has the constant solutions $u=5$ and $u=-1$ with the corresponding linear homogeneous solutions for $y$.
In general you might want to test $y=a_0+a_1x$ to find all linear solutions. $$ x^2a_1=a_1^2x^2+2a_0a_1x+a_0^2-3a_1x^2-3a_0x-5x^2 $$ and then compare coefficients to find $a_0=0$ which then leads back to the already found solutions.