Page 95 - Schaum's Outline of Differential Equations

78 LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS [CHAR 8

The Wronskian of the two solutions was found in Problem 8.6 to be -3, which is nonzero everywhere. It
follows, first from Theorem 8.3, that the two solutions are linearly independent and, then from Theorem 8.2 that the
general solution is

8.16. Find the general solution of y" - y = 0 if it is known that two solutions are

and

It was shown in both Problems 8.9 and 8.10 that these two functions are linearly independent. It follows from
Theorem 8.2 that the general solution is

x
x
x
8.17. Two solutions of y" - 2y' + y = 0 are e and 5e . Is the general solution y = c^e + c 25e l
x
We calculate

x
x
Therefore the functions e and 5e are linearly dependent (see Theorem 8.3), and we conclude from Theorem 8.2
x
that y = C]e~ x + c 25e~ is not the general solution.
1
8.18. Two solutions of are e and Is the general solution
We have

It follows, first from Theorem 8.3, that the two particular solutions are linearly independent and then from Theorem 8.2,
that the general solution is

2
1
8.19. Three solutions of /" = 0 are x , x, and 1. Is the general solution y = c^x + c 2x + c 3?
Yes. It was shown in Problems 8.11 and 8.12 that three solutions are linearly independent, so the result is
immediate from Theorem 8.3.

1
lxc
8.20. Two solutions of y"' — 6y" + 1 ly' - 6y = 0 are e and e^. Is the general solution y = c^e* + c 2e !
No. Theorem 8.2 states that the general solution of a third-order linear homogeneous differential equation is a
combination of three linearly independent solutions, not two.

8.21. Use the results of Problem 8.16 to find the general solution of

if it is known that -sin x is a particular solution.
We are given that ;y p = -sin x, and we know from Problem 8.16 that the general solution to the associated
x
homogeneous differential equation is y h = c^ + c 2e~ . It follows from Theorem 8.4 that the general solution to the
given nonhomogeneous differential equation is