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Solve the differential equation y'' + 4y = 0.
Solve the differential equation y'' + 4y = 0.
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Practice Questions
1 question
Q1
Solve the differential equation y'' + 4y = 0.
y = C1 cos(2x) + C2 sin(2x)
y = C1 e^(2x) + C2 e^(-2x)
y = C1 cos(x) + C2 sin(x)
y = C1 e^(x) + C2 e^(-x)
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The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Questions & Step-by-step Solutions
1 item
Q
Q: Solve the differential equation y'' + 4y = 0.
Solution:
The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Steps: 5
Show Steps
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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