Find the solution of the first-order linear differential equation dy/dx + y = e^

Find the solution of the first-order linear differential equation dy/dx + y = e^x.

Find the solution of the first-order linear differential equation dy/dx + y = e^x.

Step 1: Identify the differential equation: dy/dx + y = e^x.
Step 2: Recognize that this is a first-order linear differential equation.
Step 3: Find the integrating factor, which is e^(∫1 dx) = e^x.
Step 4: Multiply the entire equation by the integrating factor e^x: e^x * (dy/dx) + e^x * y = e^(2x).
Step 5: The left side can be rewritten as the derivative of (y * e^x): d/dx(y * e^x) = e^(2x).
Step 6: Integrate both sides: ∫d/dx(y * e^x) dx = ∫e^(2x) dx.
Step 7: The left side simplifies to y * e^x, and the right side integrates to (1/2)e^(2x) + C, where C is the constant of integration.
Step 8: Set the equation: y * e^x = (1/2)e^(2x) + C.
Step 9: Solve for y by dividing both sides by e^x: y = (1/2)e^(2x) * e^(-x) + Ce^(-x).
Step 10: Simplify the equation: y = (1/2)e^x + Ce^(-x).

First-order linear differential equations – These equations can be expressed in the form dy/dx + P(x)y = Q(x), where P and Q are functions of x.
Integrating factors – An integrating factor is a function used to simplify the process of solving linear differential equations, typically calculated as e^(∫P(x)dx).
General solution – The general solution of a first-order linear differential equation includes a particular solution and a constant term representing the family of solutions.