# T 2 ln t 1 y e c 1 e t 2 ln t 1 y ce t ln t 1 2 where

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t/ 2 ( ln t 1 ) y = ± e C 1 · e t/ 2 ( ln t 1 ) y = Ce t( ln t 1 )/ 2 where C is the constant ± e C 1 . 21. dy dx = e 5 x dy dx dx = e 5 x dx y = 1 5 e 5 x + C Since y = 1 when x = 0, 1 = 1 5 e 0 + C, or C = 4 5 So, y = 1 5 e 5 x + 4 5 23. dy dx = x y 2 y 2 dy = x dx y 2 dy = x dx y 3 3 = x 2 2 + C 1 y 3 = 3 2 x 2 + C 2 where C 2 is the constant 3 C 1 , y = 3 2 x 2 + C 2 1 / 3 since y = 3 when x = 2, 3 = 3 2 ( 2 ) 2 + C 2 1 / 3 3 = ( 6 + C 2 ) 1 / 3 27 = 6 + C 2 , or C 2 = 21 So, y = 3 2 x 2 + 21 1 / 3 = 3 x 2 + 42 2 1 / 3
6.2 Introduction to Differential Equations 287 25. dy dx = y 2 ( 4 x) 1 / 2 1 y 2 dy = ( 4 x) 1 / 2 dx y 2 dy = ( 4 x) 1 / 2 dx y 1 1 = 2 3 ( 4 x) 3 / 2 + C 1 1 y = 2 3 ( 4 x) 3 / 2 C 1 Since y = 2 when x = 4, 1 2 = 2 3 ( 0 ) C 1 , or C 1 = − 1 2 1 y = 2 3 ( 4 x) 3 / 2 + 1 2 = 4 ( 4 x) 3 / 2 + 3 6 y = 6 4 ( 4 x) 3 / 2 + 3 27. dy dt = y + 1 t(y 1 ) y 1 y + 1 dy = 1 t dt 1 2 y + 1 dy = 1 t dt y 2 ln | y + 1 | = ln | t | + C 1 Since y = 2 when t = 1, 2 2 ln 3 = 0 + C 1 , or C 1 = 2 ( 1 ln 3 ) y 2 ln | y + 1 | = ln | t | + 2 ( 1 ln 3 ) 29. Let V denote the value of the investment. Then, dV dt is the rate of change of V , and since this rate is equal to 7% of its size, dV dt = 0 . 07 V 31. The rate of change of p , dp dt , is jointly proportional to p and t , so dp dt = kpt where k is a negative constant of proportionality (since p is decreasing) 33. Let C denote the cost per unit x . Then, dc dx is the rate of change of C , and since this rate is a constant 60, dC dx = 60 35. Let Q denote the number of bacteria. Then, dQ dt is the rate of change of Q , and since this rate of change is proportional to Q , dQ dt = kQ where k is a positive constant of proportionality. 37. Let P denote the population. Then dP dt is the rate of change of P , and since this rate of change is the constant 500, dP dt = 500 39. Let T m = temperature of the surrounding medium T (t) = object’s temperature at time t Then, dT dt is the rate of change of T and since this rate is proportional to T m T , dT dt = k ( T m T ) 41. Let F = total number of facts and R(t) = number of facts recalled at time t. Then, dR dt is the rate of change of R and since this rate is proportional to F R , dR dt = k(F R) 43. Let N = number of people involved and P (t) = number of people implicated at time t. Then, dP dt is the rate of change of P and since this rate is proportional to (P )(N P ) ,
288 Chapter 6. Additional Topics in Integration dP dt = kP (N P ) 45. If y = Ce kx , the derivative of y is dy dx = Ce kx · k = kCe kx = ky, the given differential equation. 47. y = C 1 e x + C 2 xe x dy dx = C 1 e x + C 2 (xe x + e x ) = (C 1 + C 2 )e x + C 2 xe x d 2 y dx 2 = (C 1 + C 2 )e x + C 2 (xe x + e x ) = (C 1 + 2 C 2 )e x + C 2 xe x d 2 y dx 2 2 dy dx + y = (C 1 + 2 C 2 )e x + C 2 xe x 2 C 1 e x 2 C 2 xe x 2 C 2 e x + C 1 e x + C 2 xe x = (C 1 + 2 C 2 2 C 1 2 C 2 + C 1 )e x + (C 2 2 C 2 + C 2 )xe x = 0 · e x + 0 · xe x = 0 49. Rate revenue changes = (# barrels)(rate selling price changes). dR dt = 400 ( 98 + 0 . 04 t), where t is in months .