The general solution of the differential equation \(x\displaystyle\frac{dy}{dx} + 2y = x^{3}\), \(x \gt 0\), is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(x^{2}y = \displaystyle\frac{x^{5}}{5} + C\)
Dividing by x puts the equation in the standard linear form \(\frac{dy}{dx} + Py = Q\):
\( \displaystyle \frac{dy}{dx} + \frac{2}{x}\,y = x^{2} \)
So \(P = \frac{2}{x}\) and \(Q = x^{2}\). The integrating factor is
\( \displaystyle \text{I.F.} = e^{\int \frac{2}{x}\,dx} = e^{2\log x} = x^{2} \)
The solution is \(y \cdot \text{I.F.} = \int Q \cdot \text{I.F.}\,dx + C\):
\( \displaystyle x^{2}y = \int x^{2} \cdot x^{2}\,dx \)
\( \displaystyle x^{2}y = \frac{x^{5}}{5} + C \)
Why the other options are incorrect
Option (1) \(xy = \frac{x^{4}}{4} + C\) — Incorrect. This uses x as the integrating factor; the factor 2 in \(P = \frac{2}{x}\) makes it \(x^{2}\).
Option (2) \(x^{2}y = \frac{x^{4}}{4} + C\) — Incorrect. This integrates the unchanged right side \(x^{3}\); after dividing by x, Q is \(x^{2}\), and \(\int x^{2} \cdot x^{2}\,dx = \frac{x^{5}}{5}\).
Option (3) \(y = \frac{x^{3}}{5} + C\) — Incorrect. Dividing the correct solution by \(x^{2}\) gives \(y = \frac{x^{3}}{5} + \frac{C}{x^{2}}\); the constant term is \(\frac{C}{x^{2}}\), not C.