Summary of "How to Solve First Order Linear Differential Equations"

This tutorial explains how to solve First Order Linear Differential Equations of the form:

dy/dx + p(x)y = q(x)

where p and q are functions of x or constants.

Definition of First Order Linear Differential Equation:
- The equation is linear if the dependent variable y appears only to the first power (no powers or nonlinear functions of y).
- The coefficient of dy/dx is constant or a function of x, but not a function of y.
- Example given: dy/dx + 5y = e^(2x).
Integrating Factor Method:
- To solve such equations, an integrating factor (IF) is used.
- The integrating factor is calculated as: μ(x) = e^(∫ p(x) dx)
- Multiplying the entire differential equation by this integrating factor transforms the left-hand side into the derivative of the product μ(x) y.
- This allows the equation to be rewritten as: d/dx (μ(x) y) = μ(x) q(x)
- Integrate both sides with respect to x to find y.

Identify p(x) and q(x) from the equation dy/dx + p(x)y = q(x).
Calculate the integrating factor: μ(x) = e^(∫ p(x) dx)
Multiply the entire differential equation by μ(x): μ(x) dy/dx + μ(x) p(x) y = μ(x) q(x)
Recognize the left side as the derivative of μ(x) y: d/dx (μ(x) y) = μ(x) q(x)
Integrate both sides with respect to x: μ(x) y = ∫ μ(x) q(x) dx + C
Solve for y: y = (1/μ(x)) (∫ μ(x) q(x) dx + C)

Equation: dy/dx + 5y = e^(2x)
Steps:
- p = 5, so integrating factor: μ = e^(∫ 5 dx) = e^(5x)
- Multiply entire equation by e^(5x): e^(5x) dy/dx + 5 e^(5x) y = e^(5x) e^(2x) = e^(7x)
- Left side is: d/dx (e^(5x) y) = e^(7x)
- Integrate both sides: e^(5x) y = ∫ e^(7x) dx + C = (1/7) e^(7x) + C
- Solve for y: y = (1/7) e^(2x) + C e^(-5x)

Equation: dy/dx - y = x
Steps:
- p = -1, so integrating factor: μ = e^(∫ -1 dx) = e^(-x)
- Multiply entire equation by e^(-x): e^(-x) dy/dx - e^(-x) y = x e^(-x)
- Left side is: d/dx (e^(-x) y) = x e^(-x)
- Integrate right side using Integration by Parts:
  - Let u = x → du = dx