Tutorial 3: Lecture/Worksheet
Solve the following differential equations:
(1) 2 y
00
(2) y
00
(3) y
00
− 5 y
− 10 y
+ 4 y
0
0
0
− 3 y
+ 25 y
= 0
= 0
+ 7 y = 0
Complex numbers: z = a + ib , a = Re ( z ), b = Im ( z ).
z = a + ib and ¯ = a − ib are complex conjugates.
If the characteristic polynomial has complex roots, r
1
= a + ib and r
2
= a − ib , b > 0, then the general solution is y = e ax
( c
1 cos( bx ) + c
2 sin( bx )) c
2 y
00
+ 4 y
0
+ 7 y = 0. Try y = ue
− 2 x
. This gives u
00
+ 3 u = 0, so that u = c
1 sin( 3 x ).
The general solution to u
00
+ ω
2 u = 0 is u = c
1 cos ωx + c
2 sin ωx .
Let’s go through a couple examples, and then you can practice.
cos(
√
3 x ) +
(1) Find a particular solution of y
00
(2) Find a particular solution of y
00
− 4 y
0
− 2 y
0
+ 3 y = e
+ y
3 x
(6 + 8 x + 12 x
= 5 cos 2 x + 10 sin 2 x
2
.
).
Your turn.
(1) Find a particular solution of y
00
(2) Find a particular solution of y
00
− 3 y
0
+ 2 y
0
+ 2 y = e
− 2 x
(2 cos 3 x − (34 − 150 x ) sin 3 x ).
+ 5 y = e
− x
[(6 − 16 s ) cos 2 x − (8 + 8 x ) sin 2 x ]
1