Solve the differential equation y' = 5y + 3.

Solve 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 the left side as the derivative of a product. The left side can be rewritten as d/dx(e^(-5x) * y).
Step 6: Integrate both sides. We integrate d/dx(e^(-5x) * y) = ∫3e^(-5x) dx.
Step 7: Solve the integral on the right side. The integral of 3e^(-5x) is -3/5 * e^(-5x) + C, where C is the constant of integration.
Step 8: Write the equation from the integration. We have e^(-5x) * y = -3/5 * e^(-5x) + C.
Step 9: Solve for y. Multiply both sides by e^(5x) to isolate y: y = -3/5 + Ce^(5x).
Step 10: Rewrite the final solution. The solution is y = (3/5) + Ce^(5x).

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