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:
“You are right to demand that an artist engage his work consciously, but you confuse two different things: solving the problem and correctly posing the question.”
—Anton Pavlovich Chekhov (1860–1904)
“But then people don’t read literature in order to understand; they read it because they want to re-live the feelings and sensations which they found exciting in the past. Art can be a lot of things; but in actual practice, most of it is merely the mental equivalent of alcohol and cantharides.”
—Aldous Huxley (1894–1963)
“When the shrivelled skin of the ordinary is stuffed out with meaning, it satisfies the senses amazingly.”
—Virginia Woolf (1882–1941)
“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)