Question

If \(y=\left(x+\sqrt{1+x^2}\right)^n\), then \((1+x^2)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}\) is

• A. \(n^2y\)
• B. \(-n^2y\)
• C. \(-y\)
• D. \(2x^2y\)

Hint:

Substitution \(x=\sinh t\) simplifies expressions with \(\sqrt{1+x^2}\).

Answer 1

The Correct Option is A

Step 1: Let \(x=\sinh t\), then \(\sqrt{1+x^2}=\cosh t\).
Step 2: Hence \[ y=(\sinh t+\cosh t)^n=e^{nt}. \]
Step 3: Using \[ (1+x^2)\frac{d^2y}{dx^2}+x\frac{dy}{dx}=\frac{d^2y}{dt^2}, \] we get \[ \frac{d^2y}{dt^2}=n^2e^{nt}=n^2y. \]