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).
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