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An overview of the Picard-Lindelöf existence-uniqueness theorem for linear differential equations, including the vector nth order theorem, second order linear theorem, and higher order linear theorem. It also covers homogeneous structures, recipes for constant-coefficient linear homogeneous differential equations, superposition, and non-homogeneous structures. theorems for first and second order equations, as well as the general case for nth order equations.
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Basic Theory of Linear Differential Equations
Theorem 1 (Picard-Lindel ¨of Existence-Uniqueness)
Let the n-vector function f (x, y) be continuous for real x satisfying |x − x 0
| ≤ a
and for all vectors y in R
n satisfying ‖y − y 0
‖ ≤ b. Additionally, assume that
∂f /∂y is continuous on this domain. Then the initial value problem
y
′ = f (x, y),
y(x 0 ) = y 0
has a unique solution y(x) defined on |x − x 0 | ≤ h, satisfying ‖y − y 0 ‖ ≤ b,
for some constant h, 0 < h < a.
The unique solution can be written in terms of the Picard Iterates
yn+1(x) = y 0 +
∫ x
x 0
f (t, yn(t))dt, y 0 (x) ≡ y 0 ,
as the formula
y(x) = yn(x) + Rn(x), lim n→∞
Rn(x) = 0.
The formula means y(x) can be computed as the iterate yn(x) for large n.
Theorem 3 (Higher Order Linear Picard-Lindel ¨of Existence-Uniqueness)
Let the coefficients a 0
(x),... , a n
(x), f (x) be continuous on an interval J con-
taining x = x 0. Assume an(x) 6 = 0 on J. Let g 1 ,... , gn be constants. Then the
initial value problem
a n
(x)y
(n) (x) + · · · + a 0
(x)y = f (x),
y(x 0
) = g 1
y
′ (x 0
) = g 2
. . .
y
(n−1) (x 0 ) = gn
has a unique solution y(x) defined on J.
Theorem 4 (Homogeneous Structure 2 nd Order)
The homogeneous equation a(x)y
′′ +b(x)y
′ +c(x)y = 0 has a general solution
of the form
yh(x) = c 1 y 1 (x) + c 2 y 2 (x),
where c 1
, c 2
are arbitrary constants and y 1
(x), y 2
(x) are independent solutions.
Theorem 5 (Homogeneous Structure n th Order)
The homogeneous equation a n
(x)y
(n) +· · ·+a 0
(x)y = 0 has a general solution
of the form
y h
(x) = c 1
y 1
(x) + · · · + c n
y n
(x),
where c 1
,... , c n
are arbitrary constants and y 1
(x),... , y n
(x) are independent
solutions.
Theorem 7 (Second Order Recipe)
Let a 6 = 0, b and c be real constant. Then the general solution of
ay
′′
′
is given by the expression y = c 1
y 1
y 2
, where c 1
, c 2
are arbitrary constants
and y 1 , y 2 are two atoms constructed as outlined below from the roots of the
characteristic equation
ar
2
The characteristic equation ar
2
y
′′ → r
2 , y
′ → r, y → 1 in the differential equation ay
′′
′
Construction of Atoms for Second Order
The atom construction from the roots r 1
, r 2
of ar
2
theorem below, organized by the sign of the discriminant D = b
2 − 4 ac.
D > 0 (Real distinct roots r 1
= r 2
) y 1
= e
r 1 x , y 2
= e
r 2 x .
D = 0 (Real equal roots r 1 = r 2 ) y 1 = e
r 1 x , y 2 = xe
r 1 x .
D < 0 (Conjugate roots r 1 = r 2 = α + iβ ) y 1 = e
αx cos(βx),
y 2 = e
αx sin(βx).
Theorem 8 (Euler’s Theorem)
The atom y = x
k e
αx cos βx is a solution of ay
′′
′
if r 1 = α + iβ is a root of the characteristic equation ar
2
(r − r 1
k divides ar
2
Valid also for sin βx when β > 0. Always, β ≥ 0. For second order, only k = 1, 2 are possible.
Euler’s theorem is valid for any order differential equation: replace the equation by any
(n)
and the characteristic equation by anr
n
Theorem 10 (Superposition)
The homogeneous equation a(x)y
′′ +b(x)y
′ +c(x)y = 0 has the superposition
property:
If y 1
, y 2
are solutions and c 1
, c 2
are constants, then the combination
y(x) = c 1
y 1
(x) + c 2
y 2
(x) is a solution.
The result implies that linear combinations of solutions are also solutions.
The theorem applies as well to an nth order linear homogeneous differential equation with continuous coeffi-
cients a 0 (x),... , an(x).
The result can be extended to more than two solutions. If y 1 ,... , yk are solutions of the differential equation,
then all linear combinations of these solutions are also solutions.
The solution space of a linear homogeneous nth order linear differential equation is a subspace S of the vector
space V of all functions on the common domain J of continuity of the coefficients.
Theorem 11 (Non-Homogeneous Structure 2 nd Order)
The non-homogeneous equation a(x)y
′′
′
solution
y(x) = y h
(x) + y p
(x),
where
(x) is the general solution of the homogeneous equation
a(x)y
′′
′
(x) is a particular solution of the nonhomogeneous equation
a(x)y
′′
′
The theorem is valid for higher order equations: the general solution of the non-homogeneous equation is
y = yh + yp, where yh is the general solution of the homogeneous equation and yp is any particular solution
of the non-homogeneous equation.
An Example
For equation y
′′ − y = 10, the homogeneous equation y
′′ − y = 0 has general solution yh = c 1 e
x
−x .
Select yp = − 10 , an equilibrium solution. Then y = yh + yp = c 1 e
x
−x − 10.
