Differential Equations
Linear Differential Equations
Premium Question
Grade 12
Question:
The general solution of $\frac{dy}{dx} + \frac{y}{x} = x^3$ is:
(1) $y = \frac{x^4}{5} + Cx$
(2) $y = \frac{x^4}{5} + \frac{C}{x$
(3) $y = \frac{x^3}{4} + \frac{C}{x}$
(4) $y = \frac{x^4}{4} + \frac{C}{x}$
Step-by-Step Solution
Key Concept: This is linear in $y$ with integrating factor $e^{\int dx/x} = x$; multiply through by $x$ and recognise the left side as $\frac{d}{dx}(xy)$.
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Correct Answer: (2)