Problem 42
Find the general integral of the homogeneous equation…
Find the general integral of the homogeneous equation $(y+\sqrt{xy})\,dx=x\,dy$.
Explanation of Method
Write $y'=y/x+\sqrt{y/x}$, substitute $y=vx$, and integrate the resulting separable equation directly (a power-law integral).
Step 1: Substitute $y=vx$
For $x>0$, $y'=v+\sqrt v$ with $y=vx,\ y'=v+xv'$, giving
\[x\frac{dv}{dx}=\sqrt v.\]
Step 2: Separate and integrate
\[v^{-1/2}\,dv=\frac{dx}{x} \quad\Longrightarrow\quad 2\sqrt v=\ln|x|+C.\]
Step 3: Back-substitute $v=y/x$
Squaring, $v=\dfrac{(\ln|x|+C)^2}{4}$, so with $y=vx$,
\[\boxed{\,4y=x(\ln|x|+C)^2,\qquad x,y\ge0.}\]
Result
Final answer:
\[\boxed{\,y=\dfrac{x(\ln|x|+C)^2}{4}.}\]