3.2 Fundamental Solutions of Linear
Homogeneous Equations
Theorem 3.2.1:
Consider the initial value problem
y   p (t ) y   q (t ) y  g (t )
y (t 0 )  y 0, y (t 0 )  y 0
where p (t ), q (t ) and g (t ) are continuous on an
interval I that contains the point t 0 . Then there
is exactly one solution of this IVP, and the
solution exists throughout the interval I .
Note:
1.(Existence and Uniqueness)
Theorem 3.2.1 states that the given (IVP) has
a solution and it is unique.
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It also states that the solution exists through any
interval I containing the initial point t 0 in which
p , q and g are continuous.
Example 1: Find the longest interval in which the
(IVP)
(t 2  3t ) y   ty   (t  3) y  0,
y (1)  2, y (1)  2
has a unique solution.
Theorem 3.2.2:
If y 1 and y 2 are two solutions of the DE
y   p (t ) y   q (t ) y  0
Then the linear combination c1 y 1  c 2 y 2 is also a
solution for any values of the constants c 1 and
c 2 on an interval I that contains the point t 0 .
2
Definition: Let y 1 and y 2 be two solutions of the
y   p (t ) y   q (t ) y  0 .
The wronskian determinate (or simply the
Wronskian) of the solutions y 1 and y 2 is given
by
y1
W ( y 1, y 2 ) 
y 1
y2
 y 1 y 2  y 2 y 1.
y 2
Theorem 3.2.3:
If y 1 and y 2 are two solutions of the DE
y   p (t ) y   q (t ) y  0
and the Wronskian W ( y 1 , y 2 )  y 1 y 2  y 2 y 1
is not zero at the point t 0 where the initial
conditions y (t 0 )  y 0, y (t 0 )  y 0 .
Then there are constants c 1 and c 2 for which
y  c1 y 1  c 2 y 2 the DE and the initial conditions.
3
Theorem 3.2.4:
If y 1 and y 2 are two solutions of the DE
y   p (t ) y   q (t ) y  0
and if there is a point t 0 where the Wronskian of
y 1 and
y 2 is nonzero, then the family of
solutions
y  c1 y 1  c 2 y 2
With arbitrary coefficients c 1 and c 2 includes
every solution of the DE.
Definition: The solution
y  c1 y 1  c 2 y 2 is
called the general solution of the DE
y   p (t ) y   q (t ) y  0 .
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The solutions
y 1 and
y2
with a nonzero
Wronskian are said to form a fundamental set of
solutions of the DE.
r1 t
r t
y

e
y

e
1
Example 2: Suppose
and 2
are
2
two solutions of y   p (t ) y   q (t ) y  0 . Show
that they form a fundamental set of solutions if
r1  r2 .
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1
2
y 1  t and y 2  t 1
Example 3: Show that
form a fundamental set of solutions of
2t 2 y   3ty   y  0, t  0.
H.W. Problems 1-13, 17-20, 23-26 (Pages
151-152).
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Example 4(Q.12, page 151). Find the longest
interval in which the (IVP)
(x  2) y   y   (x  2)(tan x ) y  0,
y (3)  1, y (3)  2
has a unique solution.
Example 5(Q.18, page 151).
2 t
If the Wronskian of f and g is t e and if
f (t )  t find g (t ) .
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Example 6(Q.25, page 151)
x
y

xe
y

x
2
Is the functions 1
and
form a
fundamental set of solutions of the DE
x 2 y   x (x  2) y   (x  2) y  0, x  0.
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