Question

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

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

Hint:

For linear differential equations, ensure the coefficient of \( \frac{dy}{dx} \) is exactly \( 1 \) before isolating your \( P(x) \) and \( Q(x) \) functions.

Answer 1

The Correct Option is A

Concept: This equation is a first-order linear differential equation of the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \). The solution is found by computing the Integrating Factor (\( \text{I.F.} \)): \[ \text{I.F.} = e^{\int P(x)\,dx} \] The general solution is then given by: \[ y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.})\,dx + C \]

Step 1: Identify \( P(x) \) and compute the Integrating Factor. Comparing the given equation to the standard form: \[ P(x) = \frac{1}{x}, \quad Q(x) = x^2 \] Now calculate the Integrating Factor: \[ \text{I.F.} = e^{\int \frac{1}{x}\,dx} = e^{\ln x} = x \]

Step 2: Apply the general solution formula. Substitute the I.F. into the solution format: \[ y \cdot x = \int (x^2 \cdot x)\,dx + C \] \[ yx = \int x^3\,dx + C \]

Step 3: Integrate the right side to find the final function. Using the power rule for integration: \[ yx = \frac{x^4}{4} + C \]