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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