Consider the differential equation, $x\frac{dy}{dx}=y(\log_e y-\log_e x + 1)$, then which of the following are true?
(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is $\log_e (\frac{y}{x}) =Cx$, where C is constant of integration
(D) Its general solution is $\log_e(\frac{x}{y})= Cy$, where C is constant of integration
(E) If $y(1) = 1$, then its particular solution is $y = x$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (E) only
Given differential equation:
$x\frac{dy}{dx}=y(\log y-\log x+1)$
Rewrite:
$\frac{1}{y}\frac{dy}{dx}=\frac{\log y-\log x+1}{x}$
Substitute $u=\frac{y}{x}$, so $y=ux$ and
$\frac{dy}{dx}=u+x\frac{du}{dx}$
Substitute into the differential equation:
$x(u+x\frac{du}{dx})=ux(\log u+1)$
$xu+x^{2}\frac{du}{dx}=ux\log u+ux$
$x^{2}\frac{du}{dx}=ux\log u$
$\frac{du}{dx}=\frac{u\log u}{x}$
$\frac{du}{u\log u}=\frac{dx}{x}$
Integrate:
$\int \frac{du}{u\log u}=\int \frac{dx}{x}$
$\log (\log u)=\log x + C$
$\log \left(\log \frac{y}{x}\right)=\log x + C$
This matches option (C).
Check linearity: equation is nonlinear → (A) false.
Check homogeneity: it reduces to a function of $\frac{y}{x}$ → (B) true.
Check particular solution for $y(1)=1$:
$u=\frac{y}{x}=1$ gives $\log 1 =0$ satisfying the equation, so $y=x$ is valid → (E) true.
Final answer: (B), (C), and (E) are correct