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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