# Integrating Factor Method for First Order Differential Equations

## General Formulation

• A general form of a first-order linear differential equation is dy/dx + p(x)y = q(x), where p(x) and q(x) are any functions of x.
• For such equations, the integrating factor method is incredibly useful.
• The integrating factor is a function, denoted often as μ(x), that is determined by the coefficient p(x) of y in the differential equation.
• To find μ(x), you must take the exponential of the integral of the coefficient of y, i.e., μ(x) = e∫p(x)dx.

## Integration and Simplification Procedure

• Once the integrating factor, μ(x), is found, it is multiplied throughout the differential equation.
• The left-hand side of the equation should then simplify to the derivative of (μ(x)y), which can be checked using the product rule.
• Integrating both sides of the equation with respect to x then gives the general solution to the differential equation.

## Finding the Particular Solution

• The constant of integration, C, arising from the integration, is determined by an initial or boundary condition specified in the question.
• Substituting this condition helps to solve for C and find the particular solution to the equation.