Differential Equations
Exact Differential Equations
Premium Question
Grade 12
Question:
The general solution of $x \frac{dy}{dx} + y = x^3$ is:
(1) $y = \frac{x^3}{4} + \frac{C}{x}$
(2) $y = \frac{x^4}{4} + \frac{C}{x}$
(3) $y = \frac{x^3}{4} + Cx$
(4) $y = \frac{x^2}{4} + \frac{C}{x}$
Step-by-Step Solution
Key Concept: Notice that the left-hand side $x \frac{dy}{dx} + y$ is exactly $\frac{d}{dx}(xy)$, so the equation can be integrated directly without first finding an integrating factor.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)