Page 93 - Schaum's Outline of Differential Equations

76 LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS [CHAR 8

(b) For a first-order differential equation, (8.1) becomes
If b^x) ^ 0, we can divide through by it, in which case (8.3) takes the form

This last equation is identical to (6.1) withp(x) = a 0(x) and q(x) = 0 (x).

8.5. Find the Wronskian of the set

8.6. Find the Wronskian of the set (sin 3x, cos 3.x}.

3
2
8.7. Find the Wronskian of the set {x, x , x }.

This example shows that the Wronskian is in general a nonconstant function.

8.8. Find the Wronskian of the set {1 - x, 1 + x, 1 - 3.x}.

x
8.9. Determine whether the set {e*, e } is linearly dependent on (—00, °°).
The Wronskian of this set was found in Problem 8.5 to be -2. Since it is nonzero for at least one point in the
interval of interest (in fact, it is nonzero at every point in the interval), it follows from Theorem 8.3 that the set is
linearly independent.