Question

Find the general solution of the differential equation \( \dfrac{dy}{dx} + \dfrac{y}{x} = x^2 \).

• A. \( y = \dfrac{x^3}{4} + \dfrac{C}{x} \)
• B. \( y = \dfrac{x^3}{3} + \dfrac{C}{x} \)
• C. \( y = \dfrac{x^3}{2} + \dfrac{C}{x} \)
• D. \( y = x^3 + \dfrac{C}{x} \)

Hint:

For first-order linear differential equations, always check the form \( \frac{dy}{dx} + P(x)y = Q(x) \). The integrating factor is \( e^{\int P(x)dx} \), which converts the left-hand side into a derivative of a product.

Answer 1

The Correct Option is A

Concept: The given equation is a first-order linear differential equation of the form \[ \frac{dy}{dx} + P(x)y = Q(x) \] The integrating factor (I.F.) is \[ I.F. = e^{\int P(x)dx} \] The solution is obtained by multiplying the equation by the integrating factor.
Step 1: {Identify \(P(x)\) and \(Q(x)\).} \[ \frac{dy}{dx} + \frac{y}{x} = x^2 \] \[ P(x) = \frac{1}{x}, \quad Q(x) = x^2 \]
Step 2: {Find the integrating factor.} \[ I.F. = e^{\int \frac{1}{x} dx} \] \[ = e^{\ln x} \] \[ = x \]
Step 3: {Multiply the differential equation by the integrating factor.} \[ x\frac{dy}{dx} + y = x^3 \] Left side becomes derivative of a product: \[ \frac{d}{dx}(xy) = x^3 \]
Step 4: {Integrate both sides.} \[ \int \frac{d}{dx}(xy)\,dx = \int x^3\,dx \] \[ xy = \frac{x^4}{4} + C \]
Step 5: {Solve for \(y\).} \[ y = \frac{x^4}{4x} + \frac{C}{x} \] \[ y = \frac{x^3}{4} + \frac{C}{x} \]