What is the solution to the differential equation y' = 5y + 3?

What is the solution to the differential equation y' = 5y + 3?

Step 1: Identify the differential equation. We have y' = 5y + 3.
Step 2: Rewrite the equation in standard form. This means we want it to look like y' - 5y = 3.
Step 3: Identify the integrating factor. The integrating factor is e^(∫-5 dx) = e^(-5x).
Step 4: Multiply the entire equation by the integrating factor. This gives us e^(-5x) * y' - 5e^(-5x) * y = 3e^(-5x).
Step 5: Recognize that the left side is the derivative of (e^(-5x) * y). So we rewrite it as d/dx(e^(-5x) * y) = 3e^(-5x).
Step 6: Integrate both sides. The left side integrates to e^(-5x) * y, and the right side integrates to -3/5 * e^(-5x) + C.
Step 7: Solve for y. We get e^(-5x) * y = -3/5 * e^(-5x) + C.
Step 8: Multiply both sides by e^(5x) to isolate y. This gives us y = (3/5) + Ce^(5x).

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