04-01-2013, 11:38 AM
First-order differential equations
First-order differential.pptx (Size: 210 KB / Downloads: 31)
Basic concepts
Recall the nonhomogeneous equation
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
In this section we will learn the variation of parameters method to solve the nonhomogeneous equation. As with the method of undetermined coefficients, this procedure relies on knowing solutions to homogeneous equation.
Variation of parameters is a general method, and requires no detailed assumptions about solution form. However, certain integrals need to be evaluated, and this can present difficulties
Summary
Suppose y1, y2 are fundamental solutions to the homogeneous equation associated with the nonhomogeneous equation above, where we note that the coefficient on y'' is 1.
To find u1 and u2, we need to solve the equations