Math 333 Logarithms Worksheet With Answers Page 9
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11
12
13Solution: The characteristic equation of the corresponding homogeneous differential
2
equation is r
+ 4 = 0. The roots of this equation are
2i, so the general solution to
the homogeneous equation is
y
= c
cos 2t + c
sin 2t.
1
2
We write
y = u
(t) cos 2t + u
(t) sin 2t
1
2
and differentiate to get
y
= u
cos 2t
2u
sin 2t + u
sin 2t + 2u
cos 2t.
1
2
1
2
Now we impose the condition
u
cos 2t + u
sin 2t = 0
(1)
1
2
so that y =
2u
sin 2t + 2u
cos 2t. From this we get
1
2
y
=
2u
sin 2t
4u
cos 2t + 2u
cos 2t
4u
sin 2t.
1
2
1
2
Substituting our expressions for y and y back into the original differential equation
yields
sec 2t = y + 4y
=
2u
sin 2t
4u
cos 2t + 2u
cos 2t
4u
sin 2t
1
2
1
2
+4u
cos 2t + 4u
sin 2t
1
2
=
2u
sin 2t + 2u
sin 2t.
1
2
Combining this with equation (1), we get the system
u
cos 2t + u
sin 2t = 0
(2)
1
2
2u
sin 2t + 2u
sin 2t = sec 2t.
(3)
1
2
Multiplying the top equation by 2 sin 2t and the bottom by 2 cos 2t gives
2
2u
cos 2t sin 2t + 2u
sin
2t = 0
1
2
2
2u
cos 2t sin 2t + 2u
cos
2t = 1.
1
2
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education