文档介绍：Section
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Second Order Linear
 ¢¡£¡¥¤§¦¨¤§©  £
 £© Differential Equations
To this point we have we have considered only first order differential equations. However,
many of the most interesting differential equations involve second derivatives. Indeed,
since acceleration is the second derivative of position, Newton’s second law of motion,
F = ma, is a second order differential equation. In general, if f is a known function of
three variables, then the equation
x¨ = f(x, ˙ x, t) ()
is called a second order differential equation. If we let y =x ˙, then () may be written
as a pair of first order differential equations
x˙= y
()
y˙= f(y, x, t).
Hence moving from the study of first order differential equations to the study of second
order differential equations is analogous to moving from the study of one algebraic equation
in one unknown to the study of two algebraic equations in two unknowns. We will make
use of this fact when we consider numerical approximations to solutions of second order
equations in Section .
As was the case with first order equations, the existence of a closed form solution to a
second order differential equation and our ability to find one when it exists depends very
much on the form of the function f in (). We shall consider closed form solutions
for only