Page 94 - Schaum's Outline of Differential Equations

CHAP. 8] LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS 77

8.10. Redo Problem 8.9 by testing directly how Eq. (8.7) is satisfied.
Consider the equation

We must determine whether there exist values of c 1 and c 2, not both zero, that will satisfy (_/). Rewriting (_/), we
x
have c 2e~ = —c\& or

For any nonzero value of c 1; the left side of (2) is a constant whereas the right side is not; hence the equality in (2)
is not valid. It follows that the only solution to (2), and therefore to (_/), is Cj = c 2 = 0. Thus, the set is not linearly
dependent; rather it is linearly independent.

2
8.11. Is the set {x , x, 1} linearly dependent on (-co, co)?
The Wronskian of this set was found in Problem 8.7 to be 2.x . 3 Since it is nonzero for at least one point in
the interval of interest (in particular, at x = 3, W =54 ^ 0), it follows from Theorem 8.3 that the set is linearly
independent.

8.12. Redo Problem 8.11 by testing directly how Eq. (8.7) is satisfied.
Consider the equation

Since this equation is valid for all x only if c 1 = c 2 = c 3 = 0, the given set is linearly independent. Note that if any of
the c's were not zero, then the quadratic equation (_/) could hold for at most two values of x, the roots of the equation,
and not for all x.

8.13. Determine whether the set (1 - x, 1 + x, 1 - 3x} is linearly dependent on (—00, °°).
The Wronskian of this set was found in Problem 8.8 to be identically zero. In this case, Theorem 8.3 provides
no information, and we must test directly how Eq. (8.7) is satisfied.
Consider the equation

which can be rewritten as

This linear equation can be satisfied for all x only if both coefficients are zero. Thus,
and

Solving these equations simultaneously, we find that Cj = -2c 3, c 2 = c 3, with c 3 arbitrary. Choosing c 3 = 1 (any other
nonzero number would do), we obtain c 1 = -2, c 2 = 1, and c 3 = 1 as a set of constants, not all zero, that satisfy (_/).
Thus, the given set of functions is linearly dependent.

8.14. Redo Problem 8.13 knowing that all three functions of the given set are solutions to the differential
equation y" = 0.
The Wronskian is identically zero and all functions in the set are solutions to the same linear differential equation,
so it now follows from Theorem 8.3 that the set is linearly dependent.

8.15. Find the general solution of y" + 9y = 0 if it is known that two solutions are

and