Question

Solve: \[ \frac{dy}{dx} = y\tan x \]

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

Hint:

Useful standard integrals: \[ \int \tan x\,dx = \ln |\sec x| + C \] For separable differential equations:
• first separate variables
• integrate both sides carefully
• absorb constants after exponentiation

Answer 1

The Correct Option is A

Concept: The given differential equation is a first-order separable differential equation. A separable differential equation is of the form: \[ \frac{dy}{dx} = f(x)g(y) \] Such equations are solved by:
• separating variables involving \(y\) and \(x\)
• integrating both sides
• simplifying the obtained expression

Step 1: Given differential equation. We are given: \[ \frac{dy}{dx} = y\tan x \] We separate variables: \[ \frac{dy}{y} = \tan x \, dx \]

Step 2: Integrating both sides. Integrate both sides: \[ \int \frac{1}{y}\,dy = \int \tan x\,dx \] We know: \[ \int \frac{1}{y}\,dy = \ln |y| \] and \[ \int \tan x\,dx = \ln |\sec x| \] Therefore: \[ \ln |y| = \ln |\sec x| + C \]

Step 3: Simplifying the expression. Exponentiating both sides: \[ |y| = e^C \sec x \] Let: \[ e^C = C_1 \] Thus: \[ y = C\sec x \] where \(C\) is an arbitrary constant.

Step 4: Matching with options. The obtained solution is: \[ y = C\sec x \] which matches option (A). Final Answer: \[ \boxed{(A)\ y = C\sec x} \]