Differential Equations
Homogeneous Differential Equations
Premium Question
Grade 12
Question:
The general solution of $\frac{dy}{dx} = \frac{y}{x} + \sin\left(\frac{y}{x}\right)$ is:
(1) $\tan\left(\frac{y}{2x}\right) = Cx$
(2) $\sin\left(\frac{y}{x}\right) = Cx$
(3) $\tan\left(\frac{y}{x}\right) = Cx$
(4) $\tan\left(\frac{y}{2x}\right) = \frac{C}{x}$
Step-by-Step Solution
Key Concept: Substitute $y = vx$ to reduce this homogeneous equation to $x \frac{dv}{dx} = \sin v$, then separate variables using $\int \csc v \, dv = \ln \left|\tan \frac{v}{2}\right|$.
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Correct Answer: (1)