Non Exact Differential Equation Integrating Factor Calculator

What is an Integrating Factor?

In differential equations, an integrating factor is a function that, when multiplied by the original equation, makes it exact. An exact differential equation is one that can be written in the form:

where M and N are functions of x and y, and the partial derivatives satisfy:

If this condition is not met, the equation is non-exact. The integrating factor helps convert it into an exact equation.

How to Use This Calculator

To use the calculator, you'll need to provide:

• The differential equation in the form M(x,y)dx + N(x,y)dy = 0
• The partial derivatives ∂M/∂y and ∂N/∂x
• The integrating factor function type (linear, exponential, etc.)

The calculator will then compute the integrating factor and display the transformed exact equation.

The Formula

The general approach to finding an integrating factor involves solving:

or

where μ is the integrating factor. The solution depends on the specific form of the differential equation.

Worked Example

Consider the differential equation:

Here, M = 2xy + y² and N = x + 2xy. The partial derivatives are:

The difference is:

Assuming the integrating factor is a function of x only, we solve:

Integrating gives:

Multiplying the original equation by this integrating factor transforms it into an exact equation.