Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
13.5. Use Newton's method to ®nd a solution of
sin…x 3 ‡ 2† ˆ1=x;
with x 0 ˆ 1andh ˆ 0:001.
13.6. Approximate the following integrals using the rectangular rule, the
trapezoidal rule, and Simpson's rule, with n ˆ 2; 4; 10; 20; 50:
(a)
(b)
Z =2
0
Z
p 2
0
Z 1
e x2 sin…x 2 ‡ 1†dx;
sin…x 2 †‡3x 2
dx;
x ‡ 4
dx
(c) p :
0 2 sin 2 x
13.7 Show that the area under a parabola, as shown in Fig. 13.7, is given by
A ˆ h
3 …y 1 ‡ 4y 2 ‡ y 3 †:
13.8. Using the improved Euler's method, ®nd the value of y when x ˆ 0:2 on
the integral curve of the equation y 0 ˆ x 2 2y through the point x ˆ 0,
y ˆ 1.
13.9. Using Taylor's method, ®nd correct to four places of decimals values of y
corresponding to x ˆ 0:2 and x ˆ0:2 for the solution of the di€erential
equation
dy=dx ˆ x y 2 =10;
with the initial condition y ˆ 1 when x ˆ 0.
13.10. Using the Runge±Kutta method and h ˆ 0:1, solve
y 0 ˆ x 2 sin…y 2 †; x 0 ˆ 1 and y 0 ˆ 4:7:
13.11. Using the Runge±Kutta method and h ˆ 0:1, solve
y 0 ˆ ye x2 ; x 0 ˆ 1 and y 0 ˆ 3:
13.12. Using Taylor's method, obtain the solution of the system
y 0 ˆ x ‡ u; u 0 ˆ 1 ‡ y
with
.
y…0† ˆ1; u…0† ˆ1:
13.13. Find to four places of decimals the solution between x ˆ 0andx ˆ 0:5 of
the equations
with y ˆ u ˆ 1 when x ˆ 0.
y 0 ˆ 1
2 …y ‡ u†; u 0 ˆ 1
2 …y2 u 2 †;
479