Homework 2.3
1.
Solve
𝑑2𝑦
𝑑𝑦
2
𝑑𝑦
1
− 2 + 𝑦 = 3𝑒 4𝑥 given 𝑥 = 0, 𝑦 = − and
=4 .
𝑑𝑥 2
𝑑𝑥
3
𝑑𝑥
3
2.
Solve
𝑑2𝑦
+ 16𝑦 = 10 cos 4𝑥 given 𝑦(0) = 3 and 𝑦′(0) = 4.
𝑑𝑥 2
3.
Determine the particular solution of 𝑥̈ + 4𝑥 = cos 2𝑡 if 𝑥(0) = 1 and 𝑥̇ (0) = 2.
4.
Determine the general solution of 𝑦 ′′′ + 𝑦 ′′ − 3𝑦 = −2𝑦 − 𝑦′′.
5.
Find the particular solution of the following initial value problem: 𝑦" + 𝑦 = 10 sin 𝑡 ;
𝑦(𝜋) = 0; 𝑦′(𝜋) = 2.
6.
Solve the following. Solutions will be used when analysing solutions of models
a)
b)
1 d 2x
2
4
+ 4 x = 0, x(0) = , x '(0) = −
2
2 dt
3
3
d 2x
+ 5 x '+ 4 x = 0, x(0) = 1, x '(0) = 1
dt 2
c)
1 d 2x
+ x '+ 5 x = 0, x(0) = −2, x '(0) = 0
2 dt 2
d)
d 2x
+ 8 x '+ 16 x = 0, x(0) = −2, x '(0) = −5
dt 2