1550078481-Ordinary_Differential_Equations__Roberts_

The faitial Value Problem y' = f(x, y); y(c) = d 69

SOLUTION

a. In this instance, the function a( x) = tan x is defined and continuous on

the intervals In= ((2n-l)n/2, (2n+ l)n/2) where n is an integer and

undefined at the endpoints of the intervals. The function b( x) = sin x is

defined and continuous for all real x. So the solution to the differential

equation (37) will exist only on the intervals I n. Integrating, we find

YI(x) = efxa(t)dt = efxtantdt = e-Inlcosxl = 1 = lsecxl.

I cos x i

One often obtains YI ( x) as the absolute value of some function, as we

did in this instance. We may choose YI ( x) to be sec x or - sec x. We

select YI ( x) = sec x. Then, integrating, we find

i

x b(t) ix sin t ix 1

v(x)= -(-)dt= -dt= sintcostdt=-sin^2 x+C.

YI t sect 2

For computational convenience, we select C = 0 and obtain the following

solution of (37)

(39)

1

y(x) = YI(x)(K + v(x)) = (secx)(K +

2

sin^2 x)

1.

= K sec x +

2

( sm x) (tan x).

b. The solution to the IVP (38) is the member of the one-parameter

family (39) which satisfies the initial condition y(n/4) = v'2. Impos-

ing the initial condition, we see that K must satisfy the equation

J2 = K sec ( i) + ~ sin ( i) tan ( i) = K J2 + ( ~) ( ~) ( 1).

Solving for K , we find K = 3/4. Hence, the solution of the IVP (38) is

3secx 1

y(x) = -

4

• 
  
  
  
  • 2
  
  
  
  

(sinx)(tanx).

Since the initial condition is specified at 7r / 4 E Io = ( -?r /2, n /2) this

solution exists on the interval (-n/2,n/2).

Newton's Law of Cooling It has been shown experimentally that un-

der certain conditions the temperature of a body can be predicted by using
Newton's law of cooling which states:
"The rate of change of t he temperature of a body is proportional to
the difference between the temperature of the body and the temperature
of the surrounding medium."