Download Q2 - KFUPM AISYS

Quiz #2
Question (10 points total)
(a) Use the reduction of order method to find a second solution of the differential equation
y 00 + 16y = 0 given that y1 (x) = cos(4x) is a solution.
(b) Given that yp1 = 3e2x and yp2 = x2 + 3x are, respectively, particular solutions of the
differential equations
y 00 − 6y 0 + 5y = −9e2x and y 00 − 6y 0 + 5y = 5x2 + 3x − 16
find a particular solution of the differential equation
y 00 − 6y 0 + 5y = −10x2 − 6x + 32 + e2x .
Solution:
(a) Set y = u(x) cos(4x). So
y 0 = −4u sin(4x) + u0 cos(4x)
and
y 00 = u00 cos(4x) − 8u0 sin(4x) − 16u cos(4x).
Thus
y 00 + 16y = cos(4x)u00 − 8 sin(4x)u0 = 0
or
u00 − 8 tan(4x)u0 = 0.
Now set w = u0 to get
w0 − 8 tan(4x)w = 0.
This is linear. An integrating factor is e−8
R
tan(4x)dx
= cos2 (4x). So
d 2
cos (4x)w = 0
dx
and so
cos2 (4x)w = c.
Thus
u0 = c sec2 (4x)
and so
u = c1 tan(4x).
So
y = u(x)y1 (x) = tan(4x) cos(4x).
(b)
1
1
yp = −2(x + 3x) +
(3e2x ) = −2x2 − 6x − e2 x.
9
3
2