Question

The general solution of the differential equation \[ \frac{dy}{dx}+\frac{y}{x}=x^2 \] is

• A. \[ y=\frac13x^3+\frac{C}{x} \]
• B. \[ y=\frac14x^4+Cx \]
• C. \[ y=\frac14x^3+C \]
• D. \[ y=\frac14x^3+\frac{C}{x} \]

Hint:

For a linear differential equation \[ \frac{dy}{dx}+P(x)y=Q(x), \] use \[ \text{I.F.}=e^{\int P(x)\,dx}. \] Then \[ y\cdot \text{I.F.} = \int Q(x)\,\text{I.F.}\,dx + C. \]

Answer 1

The Correct Option is D

Concept: Linear first-order ODE: \(\frac{dy}{dx} + \frac{1}{x}y = x^2\). Integrating factor \(\mu = e^{\int (1/x)dx} = x\). Multiply and integrate.

Step 1: \(x dy/dx + y = x^3 \Rightarrow d(xy)/dx = x^3\). Integrate: \(xy = x^4/4 + C\).

Step 2: \(y = x^3/4 + C/x\).

Step 3: Write the final answer. \(\boxed{y=\frac14x^3+\frac{C}{x}}\)