Question

The substitution required to reduce the differential equation \( \frac{dy}{dx}+\sin y \cos y \sin x = \sin 2x \cos^{2}y \) to a linear differential equation in z is

• A. \( z = \tan x \)
• B. \( z = \sin 2y \)
• C. \( z = \cos y \)
• D. \( z = \tan y \)

Hint:

When you see a term like \( \sec^2 y \frac{dy}{dx} \), immediately think of \( \frac{d}{dx}(\tan y) \). This is the fastest way to identify the correct substitution in trigonometric differential equations.

Answer 1

The Correct Option is D

Concept: To reduce the given differential equation into a linear form, we look for a substitution that converts expressions involving \(y\), \(\sin y\), and \(\cos y\) into a single variable whose derivative appears naturally. ---

Step 1: Rewriting the equation.
Given: \[ \frac{dy}{dx} + \sin y \cos y \sin x = \sin 2x \cos^2 y \] Divide the entire equation by \( \cos^2 y \): \[ \frac{1}{\cos^2 y}\frac{dy}{dx} + \frac{\sin y \cos y}{\cos^2 y}\sin x = \sin 2x \] Simplify: \[ \sec^2 y \frac{dy}{dx} + \tan y \sin x = \sin 2x \] ---

Step 2: Identifying the substitution.
We recognize: \[ \frac{d}{dx}(\tan y) = \sec^2 y \frac{dy}{dx} \] So let: \[ z = \tan y \quad \Rightarrow \quad \frac{dz}{dx} = \sec^2 y \frac{dy}{dx} \] ---

Step 3: Transforming into linear form.
Substitute into the equation: \[ \frac{dz}{dx} + z \sin x = \sin 2x \] This is a standard linear differential equation in \(z\): \[ \frac{dz}{dx} + P(x)z = Q(x) \] ---

Step 4: Conclusion.
Thus, the required substitution is: \[ \boxed{z = \tan y} \]