Question

Solve: \( \frac{dy}{dx}=3x^2 \)

• A. \( y=x^3+C \)
• B. \( y=3x+C \)
• C. \( y=x^2+C \)
• D. \( y=9x+C \)

Hint:

Remember that differentiation and integration are inverse operations. If you're unsure of your integration, you can always differentiate the proposed solution to see if it matches the original derivative.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to solve a simple differential equation, which means finding the function \( y \) whose derivative with respect to \( x \) is \( 3x^2 \).

Step 2: Key Formula or Approach:
To find \( y \) from \( \frac{dy}{dx} \), we need to perform indefinite integration.
The formula for integrating power functions is:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1) \]
Where C is the constant of integration.

Step 3: Detailed Explanation:
Given differential equation:
\[ \frac{dy}{dx}=3x^2 \]
To find \( y \), integrate both sides with respect to \( x \):
\[ \int dy = \int 3x^2 dx \]
\[ y = 3 \int x^2 dx \]
Apply the power rule for integration (\( n=2 \)):
\[ y = 3 \left( \frac{x^{2+1}}{2+1} \right) + C \]
\[ y = 3 \left( \frac{x^3}{3} \right) + C \]
\[ y = x^3 + C \]
This matches option (A).

Step 4: Final Answer:
The solution to the differential equation is \( y=x^3+C \).