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:
“Science is a dynamic undertaking directed to lowering the degree of the empiricism involved in solving problems; or, if you prefer, science is a process of fabricating a web of interconnected concepts and conceptual schemes arising from experiments and observations and fruitful of further experiments and observations.”
—James Conant (1893–1978)
“When we understand that man is the only animal who must create meaning, who must open a wedge into neutral nature, we already understand the essence of love. Love is the problem of an animal who must find life, create a dialogue with nature in order to experience his own being.”
—Ernest Becker (1924–1974)
“My residence was more favorable, not only to thought, but to serious reading, than a university; and though I was beyond the range of the ordinary circulating library, I had more than ever come within the influence of those books which circulate round the world, whose sentences were first written on bark, and are now merely copied from time to time on to linen paper.”
—Henry David Thoreau (1817–1862)
“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)