Problem 40
Find the general integral of the homogeneous equation…
Find the general integral of the homogeneous equation $xy'=y\ln\dfrac{y}{x}$.
Explanation of Method
Substitute $y=vx$ to reduce the equation to a separable form involving $\ln v$, integrate with the substitution $w=\ln v-1$, and back-substitute.
Step 1: Substitute $y=vx$
Since $y'=\dfrac{y}{x}\ln\dfrac{y}{x}=v\ln v$ and $y'=v+xv'$,
\[x\frac{dv}{dx}=v\ln v-v=v(\ln v-1).\]
Step 2: Separate and integrate
With $w=\ln v-1$, $dw=dv/v$, so
\[\frac{dv}{v(\ln v-1)}=\frac{dx}{x} \quad\Longrightarrow\quad \ln|\ln v-1|=\ln|x|+C \quad\Longrightarrow\quad \ln v=1+Kx.\]
Step 3: Back-substitute $v=y/x$
\[v=e^{1+Kx} \quad\Longrightarrow\quad y=vx=exe^{Kx}.\]
Result
Final answer:
\[\boxed{\,y=exe^{Cx}\ \ \Longleftrightarrow\ \ \ln y-\ln x-Cx=1,\qquad C\in\mathbb{R}.}\]