James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1154 Chapter 17 Second-Order Differential Equations
A second-order linear differential equation has the form
1 Psxd d 2 y
dx 2
dy
1 Qsxd
dx
1 Rsxdy − Gsxd
where P, Q, R, and G are continuous functions. We saw in Section 9.1 that equations
of this type arise in the study of the motion of a spring. In Section 17.3 we will further
pursue this application as well as the application to electric circuits.
In this section we study the case where Gsxd − 0, for all x, in Equation 1. Such equations
are called homogeneous linear equations. Thus the form of a second-order linear
homogeneous differential equation is
2 Psxd d 2 y
dx 2
dy
1 Qsxd
dx 1 Rsxdy − 0
If Gsxd ± 0 for some x, Equation 1 is nonhomogeneous and is discussed in Section
17.2.
Two basic facts enable us to solve homogeneous linear equations. The first of these
says that if we know two solutions y 1 and y 2 of such an equation, then the linear combination
y − c 1 y 1 1 c 2 y 2 is also a solution.
3 Theorem If y 1 sxd and y 2 sxd are both solutions of the linear homogeneous
equation (2) and c 1 and c 2 are any constants, then the function
is also a solution of Equation 2.
ysxd − c 1 y 1 sxd 1 c 2 y 2 sxd
Proof Since y 1 and y 2 are solutions of Equation 2, we have
Psxdy0 1 1 Qsxdy 1 9 1 Rsxdy 1 − 0
and Psxdy 2 0 1 Qsxdy 2 9 1 Rsxdy 2 − 0
Therefore, using the basic rules for differentiation, we have
Psxdy0 1 Qsxdy9 1 Rsxdy
− Psxdsc 1 y 1 1 c 2 y 2 d0 1 Qsxdsc 1 y 1 1 c 2 y 2 d9 1 Rsxdsc 1 y 1 1 c 2 y 2 d
− Psxdsc 1 y 1 0 1 c 2 y 2 0d 1 Qsxdsc 1 y 1 9 1 c 2 y 2 9d 1 Rsxdsc 1 y 1 1 c 2 y 2 d
− c 1 fPsxdy 1 0 1 Qsxdy 1 9 1 Rsxdy 1 g 1 c 2 fPsxdy 2 0 1 Qsxdy 2 9 1 Rsxdy 2 g
− c 1 s0d 1 c 2 s0d − 0
Thus y − c 1 y 1 1 c 2 y 2 is a solution of Equation 2.
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