Download MATH 308 Homework 11

Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
MATH 308 Homework 11
11.1. Find the general solution of the system
−2
1
0
x =
x.
1 −2
11.2. Find the general solution of the system


1 0
0
x0 =  2 1 −2  x.
3 2
1
Express it in terms of real-valued functions.
11.3. Find a the fundamental matrix Φ(t) = eAt of the system x0 = Ax, where
5 −1
A=
.
3
1
Use it to solve the initial value problem
0
x = Ax,
x(0) =
1
1
.
11.4. Find the general solution of the system
1 −4
0
x =
.
4 −7
11.5. Find the general solution of the system
t 2 −1
e
0
x =
x+
.
3 −2
t
11.6. Find the first four terms of the power series y(t) =
value problem
(1 + x2 )y 00 − 4xy 0 + 6y = 0,
P∞
n=0
y(0) = 1,
an xn solution of the initial
y 0 (0) = 0.
11.7. Find the general solution of the differential equation
x2 y 00 − 5xy 0 + 9y = 0,
x > 0.
11.8. Show that the given differential equation has a regular singular point at x = 0.
Determine the indicial equation, its roots rP
1 , r2 , and the recurrence relation for the
n
coefficients of the series solution y(x) = xr ∞
n=0 an x .
2xy 00 + y 0 + xy = 0.
11.9. Write a recurrent formula for the values yk of approximations of the solution of the
initial value problem
y 0 = 2y − 3t,
y(0) = 1
using Euler Method with step h = 0.1. Find the values of y1 , y2 .