Question

If $y=\frac{x}{\log \left|Cx\right|}$ (where $C$ is a parameter) is the general solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+g\left(\frac{y}{x}\right)$ then the function $g\left(\frac{x}{y}\right)$ is

• A. $\frac{x^2}{y^2}$
• B. $\frac{{-x}^2}{y^2}$
• C. $\frac{y^2}{x^2}$
• D. $\frac{{-y}^2}{x^2}$

Answer 1

Answer: (D)

$\frac{{-y}^2}{x^2}$

$y=\frac{x}{\log |Cx|}⟹\frac{dy}{dx}=\frac{\log |Cx|-\frac{1}{|C|}}{(\log |Cx|{)}^2}=\frac{1}{\log |Cx|}+g(\frac{1}{\log |Cx|})⟹g\left(t\right)=\frac{-1}{t^2}⟹g(\frac{x}{y})=\frac{-y^2}{x^2}$