Differential Equations
Equations Reducible to Variable Separable
Premium Question
Grade 12
Question:
The general solution of $\frac{dy}{dx} = \sec(x + y)$ is:
(1) $\tan\left(\frac{x + y}{2}\right) = x + C$
(2) $\tan\left(\frac{x + y}{2}\right) = y + C$
(3) $\tan(x + y) = y + C$
(4) $\cot\left(\frac{x + y}{2}\right) = y + C$
Step-by-Step Solution
Key Concept: Put $u = x + y$ so that $\frac{du}{dx} = 1 + \frac{dy}{dx}$, turning the equation into the separable form $\frac{du}{1 + \sec u} = dx$; then simplify $\frac{\cos u}{1 + \cos u}$ using the half-angle identity $1 + \cos u = 2 \cos^2 \frac{u}{2}$.
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Correct Answer: (2)