Thomas Calculus 13th [Solutions]

670 Chapter 9 First-Order Differential Equations
39-42. Example CAS commands:
Mathematica: (assigned functions, step sizes, and values for initial conditions may vary)
Problems 39 - 42 involve use of code from Problems 23 - 34 together with the above code for Eulers method.
9.2 FIRST-ORDER LINEAR EQUATIONS
dy x dy
x
dx dx x x
1. x y e 1 y e
, P( x) 1 , Q( x)
1
P( x) dx ln x
P( x) dx dx ln | x | ln x, x 0 v( x)
e e x
x
x
1 1 1 x
y v( x) Q( x) dx x
e
dx e C
e C
, x 0
v( x)
x x x x
x
x
e
x
x
x dy x dy
x
dx
dx
2. e 2 e y 1 2 y e , P ( x ) 2, Q ( x ) e
P( x) dx 2x
P( x) dx 2dx 2 x v( x)
e e
2 2
y e e dx e dx e C e Ce
1 x x 1 x 1 x x x
2x e
2x e
2x
e
x
3. sin x
dy 3 sin x
2 dx x
3
x
x
3
x
xy 3 y , x 0 y , P( x) 3 , Q( x) sin x
3
3 3 ln 3
3ln ln , 0 ( )
x dx x x x v x e x
3
y 1 x sin dx 1 sin x dx 1 cos x C cos , x 0
x C x
3 3 3 3 3
x x x x x
x
x
4.
tan cos 2 , dy
2
y x y x x
2 2 dx
tan x y cos x ,
2
P( x) tan x, Q( x) cos x
1
sin
1 ln(cos x) 1
tan x dx
x
dx ln | cos x | ln (cos x) , x v( x) e (cos x) sec x
cos x
2 2
1
2
y sec x cos x dx (cos x) cos x dx (cos x) sin x C sin xcos x C cos x
sec x
dy
dy
dx x dx x x x
2
2
ln x 2
5. x 2y 1 1 , x 0 2 y 1 1 , P( x) 2 , Q( x)
1 1
2
2
2 2ln ln , 0 ( )
x dx x x x v x e x
x x x
1 2 1 1 1 1
2
x
1 1 C
2
x x 2
x
2
x
2
x 2 2 x 2
x
y x dx ( x 1) dx x C , x 0
dy
6. (1 x) y y x 1 y , P 1
dx 1 x 1 x
( x ) 1 , Q
x
( x ) 1
1 ln(1 ),
1 x dx x since ln(1 x)
x 0 v( x) e 1
x
x 3/2
3/2
(1 )
2x C
x x x x x x
y 1 x dx 1 x dx 1 2 x C
1 1 1 1 3 3(1 ) 1
x
x
7.
dy 1 1 x/2
y e P( x) 1 , 1 x/2 1
x/2
Q( x) e P( x) dx x v( x)
e
dx 2 2 2
2 2
1 x/2 1 x/2 x/2 1 x/2 1 1 x/2 x/2
y e e dx e dx e x C xe Ce
x/2
e
2 2 2 2
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