Problem 1: Section 2.2, #31
Solve the homogeneous differential equation using the substitution $y(x) = xv(x)$
$$ dy/dx = (x^2 + xy + y^2)/x^2$$
First note: Since $y(x) = xv(x)$, $dy/dx = v(x) + x(dv/dx)$. Dividing the numerator and denominator by $x^2$ and substituting $v = y/x$ yields the homogeneous equation
$$
v + x(dv/dx) = 1 + v + v^2\implies dx/x = dv/(1 + v^2).
$$
Take the integral of both sides
$$ \ln|x| + C = \arctan(v). $$
Substitute $y/x = v$
$$\arctan(y/x) - \ln|x| = C.$$
Observe how I modified the source to provide a proper formatting. Note that the last equation could be resolved with respect to $y$: $y=x\tan (C\ln |x|)$.
