Find the solution of the differential equation y' = 2y + 3.

Find the solution of the differential equation y' = 2y + 3.

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

Linear First-Order Differential Equations – These equations can be expressed in the form y' + P(x)y = Q(x) and can be solved using an integrating factor.
Integrating Factor – An integrating factor is used to solve linear differential equations and is typically of the form e^(∫P(x)dx).
General Solution – The general solution of a differential equation includes a constant (C) that represents a family of solutions.