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:
“Certainly, young children can begin to practice making letters and numbers and solving problems, but this should be done without workbooks. Young children need to learn initiative, autonomy, industry, and competence before they learn that answers can be right or wrong.”
—David Elkind (20th century)
“While I am in favor of the Government promptly enforcing the laws for the present, defending the forts and collecting the revenue, I am not in favor of a war policy with a view to the conquest of any of the slave States; except such as are needed to give us a good boundary. If Maryland attempts to go off, suppress her in order to save the Potomac and the District of Columbia. Cut a piece off of western Virginia and keep Missouri and all the Territories.”
—Rutherford Birchard Hayes (1822–1893)
“Why does he not know how to select servants? The ordinary procedure of the nineteenth century is that when a powerful and noble personage encounters a man of feeling, he kills, exiles, imprisons or so humiliates him that the other, like a fool, dies of grief.”
—Stendhal [Marie Henri Beyle] (1783–1842)
“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)