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: Let u = a/(a+bx)
From the first derivative, $xy'-y=\dfrac{ax}{a+bx}$.

Step 2: Differentiate Again
Differentiate $xy'-y$: $xy''+y'-y'=xy''=\dfrac{a(a+bx)-abx}{(a+bx)^{2}}=\dfrac{a^{2}}{(a+bx)^{2}}$.

Step 3: Multiply by x squared
$x^{3}y''=\dfrac{a^{2}x^{2}}{(a+bx)^{2}}=(xy'-y)^{2}$, option (B).

Final Answer:
The required expression is $\left(x\dfrac{dy}{dx}-y\right)^{2}$. \[ \boxed{\text{(B) }\left(x\dfrac{dy}{dx}-y\right)^{2}} \]