Download Existence of a Unique Solution

Existence of a Unique Solution
Let the coefficient functions and g(x) be
continuous on an interval I and let the leading
coefficient function not equal 0 for every x in
I. If x is any point in this interval, then a
solution y(x) of the initial-value problem exists
on the interval and is unique.
Homogeneous Equations
Nonhomogeneous Equations
dny
d n 1 y
dy
an ( x) n  an 1 ( x) n 1  ..........  a1  a0 ( x) y  0
dx
dx
dx
dny
d n 1 y
dy
an ( x) n  an 1 ( x) n 1  ..........  a1  a0 ( x) y  g ( x)
dx
dx
dx
Superposition Principle—
Homogeneous Equations
Let y1 , y2 , ...... yk be solutions of the homogeneous
nth-order differential equation on an interval
I. Then the linear combination
y  c1 y1 ( x)  c2 y2 ( x)  .......  ck yk ( x)
where the coefficients are arbitrary constants, is
also a solution on the interval.
Corollaries
• A constant multiple y  c1 y1 ( x) of a solution of
a homogeneous linear differential equation is
also a solution.
• A homogeneous linear differential equation
always possesses the trivial solution y  0 .
Linear Dependence/Independence
A set of functions f1 ( x), f 2 ( x), ....... f n ( x) is said to
be linearly dependent on an interval I if there
exist constants c1 , c2 ,.......cn , not all zero, such
that c1 f1 ( x)  c2 f 2 ( x)  ......  cn f n ( x)  0 for every x
in the interval. If the set of functions is not
linearly dependent on the interval, it is said to
be linearly independent.
Criterion for Linearly Independent
Solutions
Let y1 , y2 ,......, yn be n solutions of the
homogeneous linear nth-order differential
equation on an interval I. Then the set of
solutions is linearly independent on I if and
only if W ( y1 , y2 ,....., yn )  0 for every x in the
interval.
Fundamental Set of Solutions
Let set y1 , y2 ,......., yn of n linearly independent
solutions of the homogeneous linear nthorder differential equation on an interval I is
said to be a fundamental set of solutions on
the interval.
There exists a fundamental set of solutions for
the homogeneous linear nth-order DE on an
interval I.
General Solution-Homogeneous
Equation
Let y1 , y2 ,......., yn be a fundamental set of
solutions of the homogeneous linear nthorder differential equation on an interval I.
Then the general solution of the equation on
the interval is
y  c1 y1 ( x)  c2 y2 ( x)  .......  cn yn ( x)
where the coefficients are arbitrary constants.