Question

If y=x+√(1+x²), then (1+x²)dfracd²ydx²+x(dy)/(dx) is

• A. \(y\)
• B. \(-y\)
• C. \(2x^2y\)
• D. 0

Hint:

For expressions of the form x+√(1+x²), simplify after combining terms with a common denominator.

Answer 1

The Correct Option is A

\( y' = 1 + \dfrac{x}{\sqrt{1 + x^2}} \),
\( y'' = \dfrac{1}{(1 + x^2)^{3/2}} \)
\( (1 + x^2)y'' + x y' \)
\( = \dfrac{1}{\sqrt{1 + x^2}} + x + \dfrac{x^2}{\sqrt{1 + x^2}} \)
\( = x + \sqrt{1 + x^2} = y \)