Differential Equations Study Sheet Page 8
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
104
Second Order Differential Equations
2
d
y
dy
Form:
+ p(t)
= q(t)y = f (t).
2
dt
dt
Homogeneous if f (t) = 0.
given solutions y
and y
to the 2nd order differential equation, you must check the
1
2
Wronskian if both solutions are from real roots of the characteristic.
y
y
1
2
W = det
.
(6)
y
y
1
2
If W is equal to 0 anywhere on the interval of consideration, then y
and y
are not
1
2
linearly independent.
General solution given y
and y
is found as usual by the linearization theorem.
1
2
2
Characteristic polynomial of a 2nd order with constant coefficients: as
+ bs + c = 0.
st
Solutions of the form y(t) = e
.
2
b
b
4ac
s =
+ /
.
2a
2a
2
– if b
4ac > 0, then two distinct real roots.
2
– if b
4ac < 0, then complex roots.
2
– b
4ac = 0, then repeated real roots.
4.1
Two real distinct Roots
Two real roots, s
and s
.
1
2
s
t
s 2t
General solution = y(t) = k
e
+ k
e
.
1
1
2
4.2
Complex Roots
Complex Roots, s
= p + iq and s
= p
iq.
1
2
pt
pt
General solution = y(t) = k
e
cos(qt) + k
e
sin(qt).
1
2
4.3
Repeated Roots
Repeated Root, s
.
1
b
b
t
t
2a
a2
General solution = y(t) = k
e
+ k
te
.
1
2
8
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education