Example 1

(a)     Find the general solution of the equation y' + y cos t = 0.

(b)     Find the particular solution with initial value y(0) = ½.

(c)     Find the particular solution with initial value y(2) = ½.

SOLUTION

(a)    First evaluate the integral

∫ cos t dt = sin t + B.

General solution:

y(t) = Ce^{-sin t}.

(b)    First substitute and solve for C.

y(0) = ½ = Ce^{-sin 0} = Ce⁰ = C.

Particular solution:

y(t) = ½e^{-sin t}.

(c)    Substitute and solve for C.

y(2) = ½ = Ce^{-sin 2},

C = ½e^{sin 2} = 1.2413.

Particular solution:

y(t)= 1.2413e^{-sin t}.

The solution to this example is shown in Figure 14.2.1.

Figure 14.2.1 Example 1