The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y\log\left(\frac{y}{x}\right)$ is:
Step-by-Step Solution
Key Concept: This is a homogeneous differential equation requiring the substitution y = vx to reduce it to a separable form. After substitution and simplification, the key step involves integrating $\frac{dv}{v(\log v - 1)} = \frac{dx}{x}$ using the substitution t = log v, which yields log(t-1) = log(Cx), leading to the general solution with arbitrary constant C appearing in both additive and multiplicative forms.
For the differential equation $x\frac{dy}{dx} = y\log\left(\frac{y}{x}\right)$, substitute $y = vx$ to get $v + x\frac{dv}{dx} = v\log v$. This simplifies to $\frac{dv}{v(\log v - 1)} = \frac{dx}{x}$. Integrating both sides with substitution $t = \log v$ yields $\log(t-1) = \log(xC)$, leading to $\log\frac{y}{x} = 1 + xC$ and finally $y = xe^{1+xC} = cxe^{xC}$.
Correct Answer: 2,3