Problems illustrate-for the special case of first-order linear equations-techniques that w...

Problems illustrate-for the special case of first-order linear equations-techniques that will be important when we study higher-order linear equations in Chapter.

(a) Find constants A and B such that yp(x) = A sin x + B cos x is a solution of dy/dx + y = 2 sin x.

(b) Use the result of part (a) and the method of Problem 31 to find the general solution of dy/dx + y = 2 sin x

(c) Solve the initial value problem dy/dx + y = 2 sin x, y(0) = 1.

Problem

Problems illustrate-for the special case of first-order linear equations-techniques that will be important when we study higher-order linear equations in Chapter.

(a) Show that

is a general solution of dy/dx + P(x)y = 0.

(b) Show that

is a particular solution of dy/dx + P(x)y = Q(x).

(c) Suppose that yc(x) is any general solution of dy/dx + P(x)y = 0 and that yp(x) is any particular solution of dy/dx + P(x)y = Q(x). Show that y(x) = yc(x) + yp(x) is a general solution of dy/dx + P(x)y − Q(x).