11
Systems of Nonlinear Differential Equations
Exercises 11.1
1. The corresponding plane autonomous system is x = y, y = -9 sin x.
If (x, y) is a critical point, y = 0 and -9 sin x = 0. Therefore x = n and so the critical points are (n, 0) for n = 0, 1, 2
20
Conformal Mappings and Applications
Exercises 20.1
1. For w =
1 -y 1 1 x 1 1 and v = 2 . If y = x, u = ,u= 2 ,v=- , and so v = -u. The image is the 2 2 z x +y x +y 2 x 2 x
line v = -u (with the origin (0, 0) excluded.) 2. If y = 1, u = x 1 -1 and v = 2
19
Series and Residues
Exercises 19.1
1. 5i, 5, 5i, 5, 5i 3. 0, 2, 0, 2, 0 5. Converges. To see this write the general term as 6. Converges. To see this write the general term as 7. Converges. To see this write the general term as 8. Diverges. To see this
16
Numerical Solutions of Partial Differential Equations
Exercises 16.1
1. The figure shows the values of u(x, y) along the boundary. We need to determine u11 and u21 . The system is u21 + 2 + 0 + 0 - 4u11 = 0 1 + 2 + u11 + 0 - 4u21 = 0 or -4u11 + u21 = -
15
Integral Transform Method
Exercises 15.1
1. (a) The result follows by letting = u or u = cfw_t-1/2 =
2
2 in erf( t ) =
t
e-u du.
2
0
(b) Using
s1/2
and the first translation theorem, it follows from the convolution theorem that
t 0
1 erf( t) =
e-
14
Boundary-Value Problems in Other Coordinate Systems
Exercises 14.1
1. We have A0 = 1 2
0 0
u0 d =
u0 2
1 An = Bn = and so 1
u0 cos n d = 0 u0 sin n d =
0
u0 [1 - (-1)n ] n
u0 u0 u(r, ) = + 2 2. We have A0 = An = Bn = and so u(r, ) =
n=1
1 - (-1)n n
13
Boundary-Value Problems in Rectangular Coordinates
Exercises 13.1
1. If u = XY then ux = X Y, uy = XY , X Y = XY , and X Y = = 2 . X Y Then X 2 X = 0 so that X = A1 e x ,
2
and Y 2 Y = 0
Y = A2 e y ,
2
and u = XY = c1 ec2 (x+y) . 2. If u = XY then ux =
1
Introduction to Differential Equations
Exercises 1.1
1. Second-order; linear. 2. Third-order; nonlinear because of (dy/dx)4 . 3. The differential equation is first-order. Writing it in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y becaus
2
First-Order Differential Equations
Exercises 2.1
1.
y
2.
y y
x t
x t
3.
y
4.
y y
x
x t
5.
y
6.
y
x
x
7.
y
8.
y
x
x
17
Exercises 2.1
9.
y
10.
y
x
x
11.
y
12.
y
x
x
13.
y
14.
y
x
x
15. Writing the differential equation in the form dy/dx = y(1 - y)(1 + y)
10
Systems of Linear Differential Equations
Exercises 10.1
1. Let X =
x . Then y X = x . Then y X = 4 5 -7 0 X. 3 4 -5 8 X.
2. Let X =
x 3. Let X = y . Then z
-3 X = 6 10 x 4. Let X = y . Then z
4 -1 4
-9 0 X. 3
1 X = 1 -1 x 5. Let X = y . Then z 1 X =
9
1.
Vector Calculus
Exercises 9.1
2. 3.
4.
5.
6.
7.
8.
9.
Note: the scale is distorted in this graph. For t = 0, the graph starts at (1, 0, 1). The upper loop shown intersects the xz-plane at about (286751, 0, 286751). 10.
375
Exercises 9.1
11. x = t, y
8
Matrices
Exercises 8.1
1. 2 4 6. 8 1
2. 3 2 7. Not equal
3. 3 3 8. Not equal
4. 1 3 9. Not equal
5. 3 4 10. Not equal
11. Solving x = y - 2, y = 3x - 2 we obtain x = 2, y = 4. 12. Solving x2 = 9, y = 4x we obtain x = 3, y = 12 and x = -3, t = -12. 13. c
6
Numerical Solutions of Ordinary Differential Equations
Exercises 6.1
All tables in this chapter were constructed in a spreadsheet program which does not support subscripts. Consequently, xn and yn will be indicated as x(n) and y(n), respectively. 1.
h =
5
Series Solutions of Linear Equations
Exercises 5.1
1. lim
n
an+1 2n+1 xn+1 /(n + 1) 2n = lim = lim |x| = 2|x| n n n + 1 an 2n xn /n (-1)n converges by n n=1
The series is absolutely convergent for 2|x| < 1 or |x| < 1/2. At x = -1/2, the series
the alter
3
Higher-Order Differential Equations
Exercises 3.1
1. From y = c1 ex + c2 e-x we find y = c1 ex - c2 e-x . Then y(0) = c1 + c2 = 0, y (0) = c1 - c2 = 1 so that c1 = 1/2 and c2 = -1/2. The solution is y = 1 ex - 1 e-x . 2 2 2. From y = c1 e4x + c2 e-x we