Question

Which of the following differential equations given below has the solution y=log⁡x?

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

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

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

(d) x \(\frac{d^2 y}{dx^2}\)-log⁡x=0

The question was asked during an internship interview.

This is a very interesting question from General and Particular Solutions of Differential Equation in division Differential Equations of Mathematics – Class 12

Answer 1

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

The best I can explain: Consider the function y=log⁡x

Differentiating w.r.t x, we get

\(\frac{dy}{dx}=\frac{1}{x} \)–(1)

Differentiating (1) w.r.t x, we get

\(\frac{d^2 y}{dx^2}=-\frac{1}{x^2} \)

∴\(\frac{d^2 y}{dx^2}+(\frac{dy}{dx})^2=-\frac{1}{x^2}+(\frac{1}{x})^2\)

=-\(\frac{1}{x^2}+\frac{1}{x^2}\)=0.