Question

If $\dfrac{dy}{dx}=y\tan x$, then the solution of the differential equation is:

• A. $y=C\sin x$
• B. $y=C\cos x$
• C. $y=C\sec x$
• D. $y=C\tan x$

Hint:

Remember the standard integral: \[ \int \tan x\,dx=\log |\sec x|+C \] which is extremely common in separable differential equations.

Answer 1

The Correct Option is C

Understanding the Concept: This is a separable differential equation. We separate variables involving $y$ and $x$ on opposite sides and then integrate.
Step 1: Separate the variables.
Given: \[ \frac{dy}{dx}=y\tan x \] Rearranging: \[ \frac{dy}{y}=\tan x\,dx \]
Step 2: Integrate both sides.
Integrating: \[ \int\frac{1}{y}\,dy=\int\tan x\,dx \] \[ \log |y|=\log |\sec x|+C \]
Step 3: Simplify the solution.
Exponentiating: \[ |y|=e^C\sec x \] Let: \[ e^C=C_1 \] Thus, \[ y=C\sec x \] Hence, \[ \boxed{y=C\sec x} \]