Problem 38
Find the general integral of the homogeneous equation…
Find the general integral of the homogeneous equation $y'=\dfrac{x}{y}+\dfrac{y}{x}$.
Explanation of Method
Substitute $y=vx$ to reduce the equation to the simple separable form $v\,dv=dx/x$, then back-substitute.
Step 1: Substitute $y=vx$
Since $y'=\dfrac{1}{v}+v$ with $y=vx,\ y'=v+xv'$,
\[x\frac{dv}{dx}=\frac{1+v^2}{v}-v=\frac1v \quad\Longrightarrow\quad v\,dv=\frac{dx}{x}.\]
Step 2: Integrate and back-substitute
\[\frac{v^2}{2}=\ln|x|+C \quad\Longrightarrow\quad v^2=2\ln|x|+K \quad\Longrightarrow\quad \boxed{\,y^2-2x^2\ln|x|=Cx^2.}\]
Result
Final answer:
\[\boxed{\,y^2=x^2(C+2\ln|x|).}\]