In this exercise, we explore the connections between the method of integrating factors dis...

In this exercise, we explore the connections between the method of integrating factors discussed in this section and the Extended Linearity Principle. Consider the nonhomogeneous linear equation

where a(t) and b(t) are continuous for all t .

(a) Let

Show that μ(t) is an integrating factor for the nonhomogeneous equation.

(b) Show that 1/μ(t) is a solution to the associated homogeneous equation.

(c) Show that

is a solution to the nonhomogeneous equation.

(d) Use the Extended Linearity Principle to find the general solution of the nonhomogeneous equation.

(e) Compare your result in part (d) to the formula

for the general solution that we obtained on page.

(f) Illustrate the calculations that you did in this exercise for the example