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 Concept:
This problem asks for the solution of a first-order ordinary differential equation.
Solving a differential equation of the form \(\frac{dy}{dx} = f(x)\) involves finding the antiderivative (integral) of the function \(f(x)\).
This process is the reverse of differentiation.
When we perform indefinite integration, we must always add an arbitrary constant of integration \(C\), because the derivative of any constant is zero.
Step 2: Key Formula or Approach:
The fundamental power rule for integration states:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1) \]
The method used here is "Separation of Variables," where we group \(dy\) terms on one side and \(dx\) terms on the other.
Step 3: Detailed Explanation:
Given the equation:
\[ \frac{dy}{dx} = 3x^2 \]

Step 3.1: Separate the differentials:
Multiply both sides by \(dx\):
\[ dy = 3x^2 dx \]

Step 3.2: Integrate both sides:
Place the integral sign on both sides of the equation:
\[ \int dy = \int 3x^2 dx \]

Step 3.3: Evaluate the integrals:
The integral of \(1\) with respect to \(y\) is simply \(y\):
\[ \int 1 dy = y \]
For the right side, move the constant \(3\) outside the integral:
\[ \int 3x^2 dx = 3 \int x^2 dx \]
Apply the power rule where \(n = 2\):
\[ = 3 \times \left( \frac{x^{2+1}}{2+1} \right) + C \]
\[ = 3 \times \left( \frac{x^3}{3} \right) + C \]

Step 3.4: Simplify the expression:
The \(3\) in the numerator and the \(3\) in the denominator cancel each other out:
\[ y = x^3 + C \]
Step 4: Final Answer:
The general solution to the differential equation is \(y = x^3 + C\).