Question

Given $\frac{d^2y}{dx^2}=cot\,x\frac{dy}{dx}+4y\,cosec^2x=0$. Changing the independent variable x to z by the substitution z=log tan $\frac{x}{2}$, the equation is changed to

• A. $\frac{d^2y}{dz^2}+\frac{3}{y}=0$
• B. $2\frac{d^2y}{dz^2}+e^y=0$
• C. $\frac{d^2y}{dz^2}-4y=0$
• D. $\frac{d^2y}{dz^2}+4y=0$

Answer 1

Answer: (C)

$\frac{d^2y}{dz^2}-4y=0$

Answer 2

we need to simplify this equation in terms of z by expressing cot(x), sec^2(x/2), cosec^2(x), and their derivatives in terms of z using the given substitution z = log(tan(x/2)).

This simplification process requires rewriting these trigonometric functions in terms of tan(x/2) and then expressing tan(x/2) in terms of z.

At this point, the equation can be quite complex due to the trigonometric functions and their derivatives involved. The goal is to express everything in terms of z, simplify as much as possible, and obtain an equation involving y, z, and their derivatives.

The correct answer is option (C): $\frac{d^2y}{dz^2}-4y=0$