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Find the general solution of the equation y'' - 5y' + 6y = 0.
Find the general solution of the equation y'' - 5y' + 6y = 0.
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Practice Questions
1 question
Q1
Find the general solution of the equation y'' - 5y' + 6y = 0.
y = C1 e^(2x) + C2 e^(3x)
y = C1 e^(3x) + C2 e^(2x)
y = C1 e^(x) + C2 e^(2x)
y = C1 e^(4x) + C2 e^(5x)
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The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).
Questions & Step-by-step Solutions
1 item
Q
Q: Find the general solution of the equation y'' - 5y' + 6y = 0.
Solution:
The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).
Steps: 6
Show Steps
Step 1: Write down the given differential equation: y'' - 5y' + 6y = 0.
Step 2: Identify the characteristic equation by replacing y'' with r^2, y' with r, and y with 1. This gives us r^2 - 5r + 6 = 0.
Step 3: Factor the characteristic equation r^2 - 5r + 6. Look for two numbers that multiply to 6 and add to -5. The numbers are -2 and -3.
Step 4: Write the factored form: (r - 2)(r - 3) = 0.
Step 5: Set each factor equal to zero to find the roots: r - 2 = 0 gives r = 2, and r - 3 = 0 gives r = 3.
Step 6: Write the general solution using the roots found. The general solution is y = C1 e^(2x) + C2 e^(3x), where C1 and C2 are constants.
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