Page 76 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 59

This differential equation is linear. Its solution is given in Problem 6.15 as

(0)
Since T = 50 when t = 0,ii follows from (1) that 50 = ce~* + 100, or c = -50. Substituting this value into (1),
we obtain

5k
Ait =5, we are given that T= 60; hence, from (2), 60 = —5Qe + 100. Solving for k, we obtain

Substituting this value into (2), we obtain the temperature of the body at any time t as

(a) We require t when T = 75. Substituting T=75 into (3), we have

Solving for t, we find

(b) We require T when t = 20. Substituting t = 20 into (3) and then solving for T, we find

7.10. A body at an unknown temperature is placed in a room which is held at a constant temperature of 30° F.
If after 10 minutes the temperature of the body is 0° F and after 20 minutes the temperature of the body
is 15° F, find the unknown initial temperature.
From (7.2),

Solving, we obtain

At t= 10, we are given that T=0. Hence, from (_/),

At t= 20, we are given that T= 15. Hence, from (_/) again,

Solving (2) and (3) for k and c, we find