Question

If y^2 + xy + x^2 – 2x = 0 then ^d^2y⁄dx^2 =?

(a) \((2y+x) \frac{d^2 y}{dx^2}+(\frac{dy}{dx})^2+2 \frac{dy}{dx}+2=0\)

(b) \((2y+x) \frac{d^2 y}{dx^2}+2(\frac{dy}{dx})^2+\frac{dy}{dx}+2=0\)

(c) \((2y+x) \frac{d^2 y}{dx^2}+2(\frac{dy}{dx})^2+2 \frac{dy}{dx}+2=0\)

(d) \(x \frac{d^2 y}{dx^2}+2(\frac{dy}{dx})^2+2 \frac{dy}{dx}+2=0\)

The question was asked in exam.

The query is from Implicit Differentiation in section Partial Differentiation of Engineering Mathematics

Answer 1

Right answer is (c) \((2y+x) \frac{d^2 y}{dx^2}+2(\frac{dy}{dx})^2+2 \frac{dy}{dx}+2=0\)

Explanation: \(y^2+xy+x^2-2x=0\)

\(2y \frac{dy}{dx}+x \frac{dy}{dx}+y+2x-2=0\)

\(\frac{2yd^2 y}{dx^2}+2(\frac{dy}{dx})^2+x (\frac{d^2 y}{dx^2})+\frac{dy}{dx}+\frac{dy}{dx}+2=0\)

\((2y+x) \frac{d^2 y}{dx^2}+2(\frac{dy}{dx})^2+2 \frac{dy}{dx}+2=0\)