Question

The general solution of the differential equation \( \frac{dy}{dx} = 3x^2 \) is:

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

Hint:

When solving simple differential equations like \( \frac{dy}{dx} = f(x) \), integrate both sides with respect to \( x \) and add the constant of integration \( C \).

Answer 1

The Correct Option is A

The differential equation \( \frac{dy}{dx} = 3x^2 \) is provided, and the objective is to determine its general solution. Step 1: Integrate both sides To isolate \( y \), integrate both sides of the equation with respect to \( x \): \[ y = \int 3x^2 \, dx \] Step 2: Execute the integration The integral of \( x^2 \) is \( \frac{x^3}{3} \). Therefore: \[ y = 3 \times \frac{x^3}{3} + C = x^3 + C \] \( C \) represents the constant of integration. Answer: The general solution to the differential equation is \( y = x^3 + C \), corresponding to option (1).