Non-separable (non-homogeneous) First-order Linear Ordinary Differential Equations
First-order linear non-homogeneous ODEs (ordinary differential equations) are not separable. They can be solved by the following approach, known as an integrating factor method. Consider first-order linear ODEs of the general form:
The method for solving this equation relies on a special integrating factor, μ:
We choose this integrating factor because it has the special property that its derivative is itself times the function we are integrating, that is:
Multiply both sides of the original differential equation by μ to get:
Because of the special μ we picked, we may substitute dμ/dx for μ p(x), simplifying the equation to:
Using the product rule in reverse, we get:
Integrating both sides:
Finally, to solve for y we divide both sides by :
Since μ is a function of x, we cannot simplify any further directly.
Read more about this topic: Examples Of Differential Equations
Famous quotes containing the words ordinary and/or differential:
“The ordinary man—we have to face it: it is every bit as true of the ordinary Englishman as of the ordinary American—is an Anarchist. He wants to do as he likes. He may want his neighbor to be governed, but he himself doesn’t want to be governed.”
—George Bernard Shaw (1856–1950)
“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)