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Find the general solution of the differential equation dy/dx = 3x^2.

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Question: Find the general solution of the differential equation dy/dx = 3x^2.

Options:

  1. y = x^3 + C
  2. y = 3x^3 + C
  3. y = x^2 + C
  4. y = 3x^2 + C

Correct Answer: y = x^3 + C

Solution: 

Integrating both sides gives y = (3/3)x^3 + C = x^3 + C.

Find the general solution of the differential equation dy/dx = 3x^2.

Practice Questions

Q1
Find the general solution of the differential equation dy/dx = 3x^2.
  1. y = x^3 + C
  2. y = 3x^3 + C
  3. y = x^2 + C
  4. y = 3x^2 + C

Questions & Step-by-Step Solutions

Find the general solution of the differential equation dy/dx = 3x^2.
  • Step 1: Start with the given differential equation: dy/dx = 3x^2.
  • Step 2: To find y, we need to integrate the right side (3x^2) with respect to x.
  • Step 3: The integral of 3x^2 is (3/3)x^3, which simplifies to x^3.
  • Step 4: When we integrate, we also add a constant C to represent any constant value that could be added to y.
  • Step 5: Therefore, the general solution is y = x^3 + C.
  • Separation of Variables – The question tests the ability to integrate a simple first-order differential equation.
  • Integration – The solution requires knowledge of basic integration techniques.
  • General Solution – Understanding the concept of a general solution and the role of the constant of integration (C).
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