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Solve the differential equation y'' + 4y = 0.

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Question: Solve the differential equation y\'\' + 4y = 0.

Options:

  1. y = C1 cos(2x) + C2 sin(2x)
  2. y = C1 e^(2x) + C2 e^(-2x)
  3. y = C1 cos(x) + C2 sin(x)
  4. y = C1 e^(x) + C2 e^(-x)

Correct Answer: y = C1 cos(2x) + C2 sin(2x)

Solution: 

The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).

Solve the differential equation y'' + 4y = 0.

Practice Questions

Q1
Solve the differential equation y'' + 4y = 0.
  1. y = C1 cos(2x) + C2 sin(2x)
  2. y = C1 e^(2x) + C2 e^(-2x)
  3. y = C1 cos(x) + C2 sin(x)
  4. y = C1 e^(x) + C2 e^(-x)

Questions & Step-by-Step Solutions

Solve the differential equation y'' + 4y = 0.
Correct Answer: y = C1 cos(2x) + C2 sin(2x)
  • Step 1: Write down the differential equation: y'' + 4y = 0.
  • Step 2: Identify the characteristic equation by replacing y'' with r^2 and y with 1: r^2 + 4 = 0.
  • Step 3: Solve the characteristic equation for r: r^2 = -4.
  • Step 4: Take the square root of both sides: r = ±2i (this means the roots are complex).
  • Step 5: Use the complex roots to write the general solution: y = C1 cos(2x) + C2 sin(2x), where C1 and C2 are constants.
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