Existence and Uniqueness Theorem
Let the following equation,
be the order linear DE, where and are cont. real val. functions on a real int., and
Let be any part of and be any arbitrary real const. Then, a unique solution of (I) such that,
and the solution is defined over the entire int. .
Theorem: Superposition Principle
Let , be diff non-homog. linear DE of order , where
for any , are cont. funcs. of defined on . Let be a particular soln. of (2). Then, is
a particular soln. of
Proof
Given, is a particular soln. of (2).
To prove that is a soln. of
We have,
Replace by .
Hence, is a soln of .
First Order Linear DE
Case 1: ,
Theorem
Consider . where is a complex constant. If , then the func. is defined by is a solution, and moreover, every soln. has this form.
Proof
Let is is a soln of .
Suppose is a soln of (1).
Conversely, suppose , then
Case 2:
Theorem
Consider the DE, , where , and is a cont. func on the int. . If is a point in and is any const., then the func. defined by:
of the given DE and every soln is of this form.
Proof
Let be any soln. of .
Problems
Case 3:
Theorem
If and are cont. funcs. Let be any cont. func, where . Then the func. given by
is the soln of the eqn. on . The func. given by, (1)
homog. eqn., .
If is any constant, then is a solution of (1) and every soln. of (1) has the same form.