Download Theory of n-th Order Linear Differential Equations

Theory of n-th Order Linear Differential Equations
Uniqueness and Existence Theorem
Given the general linear n-th order initial value problem
dn−1 y
dy
dn y
+
P
(x)
+ · · · + P1 (x) + P0 (x)y = R(x) ,
n−1
n
n−1
dx
dx
dx
subject to y(x0 ) = y0 , y 0 (x0 ) = y1 , · · · , y (n−1) (x0 ) = yn−1 , where Pn−1 (x), Pn−2 (x), · · · , P0 (x)
and R(x) are functions of x, continuous on a < x < b, and a < x0 < b and
y0 , y1 , · · · , yn−1 are given real numbers, then there exists a unique solution
y(x).
General Solution of a Homogeneous Linear Equation
Let {y1 (x), y2 (x), · · · , yn (x)} be a linearly independent set of solutions to
the homogeneous n-th order linear equation
dn y
dn−1 y
dy
+
P
(x)
+ · · · + P1 (x) + P0 (x)y = 0
n−1
n
n−1
dx
dx
dx
for a < x < b. Suppose further that each Pj (x) is continuous on a < x < b.
If φ(x) is any solution to the homogeneous linear equation above, then there
exist constants d1 , d2 , · · · , dn such that
φ(x) = d1 y1 (x) + d2 y2 (x) + · · · dn yn (x) .
Hence, we say that {y1 (x), y2 (x), · · · , yn (x)} is a fundamental solution set of
the homogeneous linear differential equation, and that the general solution
is
y(x) = c1 y1 (x) + c2 y2 (x) + · · · + cn yn (x) .
where c1 , c2 , · · · , cn are arbitrary contants.
Goal : Given an n-th order linear differential equation, find n linearly independent solutions.
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General Solution of a Nonhomogeneous Equation
Let yp (x) be a particular solution of the nonhomogeneous n-th order linear
differential equation
dn y
dn−1 y
dy
+
P
(x)
+ · · · + P1 (x) + P0 (x)y = R(x)
n−1
n
n−1
dx
dx
dx
on a < x < b, where each Pj (x) and R(x) are continuous on that interval,
and let yc (x) = c1 y1 (x)+c2 y2 (x)+· · ·+cn yn (x) be a solution of the associated
homogeneous equation (the one where R(x) = 0). Then, the general solution
to the nonhomogeneous equation is
y(x) = yc (x) + yp (x) = c1 y1 (x) + c2 y2 (x) + · · · cn yn (x) + yp (x)
where c1 , c2 , · · · , cn are arbitrary contants. yc (x) is called the complementary solution, or complementary function. Thus, the general solution to the
equation is a combination of the complementary and particular solutions.
Goal: Given an n-th order linear nonhomogeneous differential equation, find
n linearly independent solutions to the corresponding homogeneous equation,
and find one particular solution of the nonhomogeneous equation.
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