Question

Find the solution of the differential equation: \[ \frac{dy}{dx} = 3x^2 \]

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

Hint:

A simple differential equation of the form \( \frac{dy}{dx} = f(x) \) is just an integration problem in disguise.
Simply integrate the right-hand side function directly to find \( y \).
Do not forget to add the constant of integration \( C \) to represent the family of solutions.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This problem asks us to solve a first-order ordinary differential equation of the form \( \frac{dy}{dx} = f(x) \).
To find the general solution, we must determine the function \( y(x) \) whose derivative with respect to \( x \) is \( 3x^2 \).

Step 2: Key Formula or Approach:
We can solve this first-order differential equation using the variable separable method.
We separate the variables \( y \) and \( x \) to opposite sides of the equation:
\[ dy = f(x) \, dx \]
Then, we integrate both sides to obtain the general solution:
\[ \int 1 \, dy = \int f(x) \, dx \]
The integration of the right side will require the standard power rule of integration:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Step 3: Detailed Explanation:
Given the differential equation:
\[ \frac{dy}{dx} = 3x^2 \]
We separate the differentials by multiplying both sides by \( dx \):
\[ dy = 3x^2 \, dx \]
Now, we integrate both sides of the equation:
\[ \int dy = \int 3x^2 \, dx \] Integrating the left-hand side with respect to \( y \):
\[ \int dy = y \] Integrating the right-hand side with respect to \( x \):
\[ \int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} + C = 3 \cdot \frac{x^3}{3} + C = x^3 + C \]
Combining both sides, we get the general solution:
\[ y = x^3 + C \]
Let us examine the incorrect options:
- Option (B) \( y = 3x^3 + C \) is incorrect because the coefficient \( 3 \) was not divided by the new exponent \( 3 \).
- Option (C) \( y = x^2 + C \) is incorrect because it represents the derivative of the right-hand side, not the integral.
- Option (D) \( y = 9x + C \) is incorrect because it is totally unrelated and does not represent the antiderivative.

Step 4: Final Answer:
The general solution of the given differential equation is \( y = x^3 + C \), which matches Option (A).