Inexact differential equation

An inexact differential equation is a differential equation of the form (see also: inexact differential)

The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739.[1]

Solution method

In order to solve the equation, we need to transform it into an exact differential equation. In order to do that, we need to find an integrating factor to multiply the equation by. We'll start with the equation itself. , so we get . We will require to satisfy . We get

After simplifying we get

Since this is a partial differential equation, it is mostly extremely hard to solve, however in some cases we will get either or , in which case we only need to find with a first-order linear differential equation or a separable differential equation, and as such either

or

References

  1. "History of differential equations – Hmolpedia". www.eoht.info. Retrieved 2016-10-16.

Further reading

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