Question

The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ is:

• A. $y x^2 = \frac{x^4}{4} + C$
• B. $y x = \frac{x^3}{3} + C$
• C. $y x^2 = \frac{x^3}{3} + C$
• D. $y x = \frac{x^4}{4} + C$

Hint:

Differentiating the LHS of the solution $y x^2$ yields $x^2 \frac{dy}{dx} + 2xy$, which is precisely $x$ times the LHS of the original differential equation. This is a quick way to verify the integrating factor.

Answer 1

The Correct Option is A

Step 1: Concept
We write the linear differential equation in standard form $\frac{dy}{dx} + P(x)y = Q(x)$ and compute the Integrating Factor $\text{I.F.} = e^{\int P(x) \, dx}$.

Step 2: Meaning
We divide the entire equation by $x$ to find $P(x)$ and $Q(x)$.

Step 3: Analysis
Dividing by $x$: \[ \frac{dy}{dx} + \left(\frac{2}{x}\right)y = x \] Here, $P(x) = \frac{2}{x}$ and $Q(x) = x$. Compute the integrating factor: \[ \text{I.F.} = e^{\int \frac{2}{x} \, dx} = e^{2\ln|x|} = e^{\ln x^2} = x^2 \] Now write the general solution: \[ y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) \, dx + C \] \[ y \cdot x^2 = \int (x \cdot x^2) \, dx + C \] \[ y x^2 = \int x^3 \, dx + C \] \[ y x^2 = \frac{x^4}{4} + C \]

Step 4: Conclusion
The solution of the differential equation is $y x^2 = \frac{x^4}{4} + C$.

Final Answer: (A)