1. x(t)=2e2t + 4e−t cos(3t) - 2e −t sin(3t), y(t)=5e2t + e−t cos(3t)+7e−t

1.
x(t) = 2e2t + 4e−t cos(3t) − 2e−t sin(3t), y(t) = 5e2t + e−t cos(3t) + 7e−t sin(3t).
"
cos(t)
− sin(t)
#
2.
~x(t) = t
3.
y(x) = 2 + 3x − x2 − 56 x3 + . . .
dx
4. dt
5.
= (x − 1)(x − 2)(x − 3), limt→∞ x(t) = 2
u(x, t) = − 12 γx2 +
6. (a)
y(t) =
12 4t
7 e
1
2
− α + β x+α+2
+ 97 e−3t − 2e2t .
(b)
P∞
an
1 < x(0) < 3.
n
γ
n=1
n3 π 3
[(−1)n − 1] +
2
nπ
y(t) = c1 t−2 + c2 t−2 ln(t).
7. (a)
ut =
1
4 uxx ,
0 < x < π, t > 0,
u(0, t) =
10,
t > 0,
ux (π, t) =
0,
t > 0,
u(x, 0) = sin(3x/2) + 10, 0 < x < π.
(b)
u(x, t) = e−9t/16 sin( 3x
2 ) + 10.
o
2 π2 t
[α − (−1)n β] e−n
sin(nπx)