Thomas Calculus 13th [Solutions]

664 Chapter 9 First-Order Differential Equations
y3
( .39794)
.39794 (.5) .34195,
2
2
y4
( .34195)
.34195 (.5) .30497,
2.5
2
y5 .27812, y6 .25745, y7 .24088, y8
.2272;
dy dx 1 2 x C;
y(1) 1 1 2 C C 1 y 1 y(5) 1 .2880
2
y x y
1 2 x
1 2 5
20. y
1
1 1 (0 sin1)
3
1,
y 1 1
2 1 sin1
3 3
1.09350,
y
2 1
3 1.09350 sin1.09350
3 3
1.29089,
y
3 1
4 1.29089 sin1.29089
3 3
1.61125,
y
4 1
5 1.61125 sin1.61125
3 3
2.05533,
y
5 1
6 2.05533 sin 2.05533
3 3
2.54694;
1 2 1 2
x C x
2 2
1 2
y xsin y csc y dy x dx ln csc y cot y x C csc y cot y e Ce
2
1 2 1 2
x
2
y x
2
1 cos y
Ce cot Ce ;
sin y
2
1
2
2
1 0
y
y(0) 1 cot Ce C cot cot 1 e
2 2 2
1 1 x
1 1 2
y 2cot cot e , y(2) 2cot cot e 2.65591
2 2
1
2
x
2
x x x x
y 1 x 1 x0 y0 e y( x0 ) 1 x0 1 x0 y0 e 1 x0 1 x0 y0 (1) y0
21.
0 0 0
dy x x
x x dy dy
1 1 0 0 1 1
dx 0 0 dx dx
x y e
0
y x x y e
0
x x y
22.
x
y f ( x), y( x0 ) y0
y f ( t) dt C,
x
0
y 0
( x x
x
0 ) ( ) 0 ( )
x
f t dt C C C y y x
f t dt y 0
0 0
23-34. Example CAS commands:
Maple:
ode : diff ( y(x), x ) y(x);
icA : [0,1];
icB : [0, 2];
icC : [0,-1];
DEplot( ode, y(x), x 0..2, [icA,icB,icC], arrows slim, linecolor blue, title "#
23 (Section 9.1)" );
Mathematica:
To plot vector fields, you must begin by loading a graphics package.
Graphics`PlotField`
To control lengths and appearance of vectors, select the Help browser, type PlotVectorField and select Go.
Clear[x, y, f ]
yprime y (2 y);
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