Find the general solution of the equation y' = 5y + 3.

Find the general solution of the equation y' = 5y + 3.

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.

First-Order Linear Differential Equations – The question tests the understanding of solving first-order linear differential equations using the integrating factor method.
Integrating Factor Method – The solution requires knowledge of how to find and apply the integrating factor to solve the differential equation.
General Solution – The question assesses the ability to derive the general solution from the particular solution obtained through integration.