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Find the general solution of the equation y' = 5y + 3.
Find the general solution of the equation y' = 5y + 3.
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Practice Questions
1 question
Q1
Find the general solution of the equation y' = 5y + 3.
y = Ce^(5x) - 3/5
y = Ce^(5x) + 3/5
y = 3/5 + Ce^(-5x)
y = 5x + C
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The integrating factor method gives the general solution y = Ce^(5x) - 3/5.
Questions & Step-by-step Solutions
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Q
Q: Find the general solution of the equation y' = 5y + 3.
Solution:
The integrating factor method gives the general solution y = Ce^(5x) - 3/5.
Steps: 10
Show Steps
Step 1: Write the differential equation in standard form: y' - 5y = 3.
Step 2: Identify the coefficient of y, which is -5.
Step 3: Calculate the integrating factor, which is e^(∫-5 dx) = e^(-5x).
Step 4: Multiply the entire equation by the integrating factor: e^(-5x) * (y' - 5y) = 3 * e^(-5x).
Step 5: The left side simplifies to the derivative of (e^(-5x) * y).
Step 6: Rewrite the equation as d/dx(e^(-5x) * y) = 3 * e^(-5x).
Step 7: Integrate both sides: ∫d/dx(e^(-5x) * y) dx = ∫3 * e^(-5x) dx.
Step 8: The left side becomes e^(-5x) * y, and the right side integrates to -3/5 * e^(-5x) + C.
Step 9: Solve for y: y = e^(5x) * (-3/5 * e^(-5x) + C).
Step 10: Simplify to get the general solution: y = Ce^(5x) - 3/5.
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