Question

If \((\mathrm{a}+\mathrm{b} x) \mathrm{e}^{\frac{y}{x}}=x\), then \(x^3 \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}\) is equal to

• A. \((y\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}-x)^{2}\)
• B. \((x\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}-y)^{2}\)
• C. \((x\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}+y)^{2}\)
• D. \((y\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}+x)^{2}\)

Hint:

Take logs to express y in terms of x, then find y' and y'' and compare with the options.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
From \((a+bx)e^{y/x}=x\): \(\dfrac{y}{x}=\log x-\log(a+bx)\).

Step 2: First Derivative
\(y=x\log x-x\log(a+bx)\).
\(y'=\log\dfrac{x}{a+bx}+1-\dfrac{bx}{a+bx}=\dfrac{y}{x}+\dfrac{a}{a+bx}\).
So \(xy'-y=\dfrac{ax}{a+bx}\).

Step 3: Second Derivative
\(y''=\dfrac{1}{x}-\dfrac{b}{a+bx}-\dfrac{ab}{(a+bx)^{2}}=\dfrac{a^{2}}{x(a+bx)^{2}}\).
So \(x^{3}y''=\dfrac{a^{2}x^{2}}{(a+bx)^{2}}\).

Step 4: Compare
This equals \(\left(\dfrac{ax}{a+bx}\right)^{2}=\left(x\dfrac{dy}{dx}-y\right)^{2}\), option (B).

Final Answer:
\(x^{3}\dfrac{d^{2}y}{dx^{2}}\) equals \(\left(x\dfrac{dy}{dx}-y\right)^{2}\). \[ \boxed{\text{(B) }\left(x\dfrac{dy}{dx}-y\right)^{2}} \]