Find the solution of the differential equation y' = 3y + 6.

Find the solution of the differential equation y' = 3y + 6.

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

Linear First-Order Differential Equations – These equations can be solved using an integrating factor, which simplifies the process of finding the general solution.
Integrating Factor – The integrating factor is a function used to multiply the differential equation to make it easier to solve.
General Solution – The general solution includes a constant (C) that represents the family of solutions to the differential equation.