# Integrating factor method for first order differential equations

# Integrating Factor Method for First Order Differential Equations

## General Formulation

- A general form of a first-order linear differential equation is
, where p(x) and q(x) are any functions of x.*dy/dx + p(x)y = q(x)* - For such equations, the
is incredibly useful.*integrating factor method* - 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.

## Understanding Your Solution

- It’s important to understand what your solution means. While it might be a strain in physics, it could represent a population in a biological model.
- Evaluate the solution at different points to predict specific behaviours of the system.
- Plotting a graph of the solution can also give an
**insight into the overall behaviour**of the system under consideration. - Carefully consider the
**units or dimensions**of your variables and parameters - consistency can be a good check for the correctness of a solution.

## Common Pitfalls and Cautions

- Ensure to multipIy the
equation by the integrating factor, not just one side.*entire* - Always double-check your calculations - especially the integral calculation to find the integrating factor.
- Be careful with your integration step, the integral of the RHS will not always be a neat or standard formula.
- Remember to solve for C using any given initial or boundary conditions to find the particular solution. Be mindful of whether the condition is an initial condition (at t=0) or a boundary condition (at a specified x value).
- Always pose and answer the question: Does my solution make sense in the given real-world context? If not, you may need to review your steps.