Use in Solving First Order Linear Ordinary Differential Equations
Integrating factors are useful for solving ordinary differential equations that can be expressed in the form
The basic idea is to find some function, called the "integrating factor," which we can multiply through our DE in order to bring the left-hand side under a common derivative. For the canonical first-order, linear differential equation shown above, our integrating factor is chosen to be
We see that multiplying through by gives
By applying the product rule in reverse, we see that the left-hand side can be expressed as a single derivative in
We use this fact to simplify our expression to
We then integrate both sides with respect to, obtaining
Finally, we can move the exponential to the right-hand side to find a general solution to our ODE:
In the case of a homogeneous differential equation, in which, we find that
where is a constant.
Read more about this topic: Integrating Factor
Famous quotes containing the words solving, order, ordinary and/or differential:
“There are horrible people who, instead of solving a problem, tangle it up and make it harder to solve for anyone who wants to deal with it. Whoever does not know how to hit the nail on the head should be asked not to hit it at all.”
—Friedrich Nietzsche (1844–1900)
“I write poetry in order to live more fully.”
—Judith Rodriguez (b. 1936)
“Even in ordinary speech we call a person unreasonable whose outlook is narrow, who is conscious of one thing only at a time, and who is consequently the prey of his own caprice, whilst we describe a person as reasonable whose outlook is comprehensive, who is capable of looking at more than one side of a question and of grasping a number of details as parts of a whole.”
—G. Dawes Hicks (1862–1941)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)