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

Step 1: Look at the equation.
\[ \frac{dy}{dx}+\sin y\cos y\sin x=\sin2x\cos^2y \] We want a substitution that turns the $y$ parts into one new variable.
Step 2: Divide through by $\cos^2y$.
\[ \frac{1}{\cos^2y}\frac{dy}{dx}+\frac{\sin y\cos y}{\cos^2y}\sin x=\sin2x \]
Step 3: Simplify each term.
Since $\frac{1}{\cos^2y}=\sec^2y$ and $\frac{\sin y\cos y}{\cos^2y}=\tan y$: \[ \sec^2y\frac{dy}{dx}+\tan y\sin x=\sin2x \]
Step 4: Spot the derivative.
Notice that $\frac{d}{dx}(\tan y)=\sec^2y\frac{dy}{dx}$. This points to the right substitution.
Step 5: Set $z=\tan y$.
Then $\frac{dz}{dx}=\sec^2y\frac{dy}{dx}$.
Step 6: Confirm the linear form.
Substituting gives \[ \frac{dz}{dx}+z\sin x=\sin2x \] which is linear in $z$. So the needed substitution is \[ \boxed{z=\tan y} \]