Lecture Notes in Differential Equations - Bruce E. Shapiro

139
The second order linear initial value problem is then
Ly = 0
y(t 0 ) = y 0
y ′ (t 0 ) = y 1
⎫
⎪⎬
⎪ ⎭
(16.42)
This is not the only way to express the constraints upon the solution. It
is also possible to have boundary conditions, of which several types are
possible:
y(t 0 ) = y 0 , y(t 1 ) = y 1 (Dirichlet Boundary Conditions) (16.43)
y(t 0 ) = y 0 , y ′ (t 1 ) = y 1 (Mixed Boundary Condition) (16.44)
y ′ (t 0 ) = y 0 , y ′ (t 1 ) = y 1 (Neumann Boundary Conditions) (16.45)
Differential equations combined with boundary conditions are called Boundary
Value Problems. Boundary Value Problems are considerably more
complex than Initial Value Problems and we will not study them in this
class.
Definition 16.6. The homogeneous linear second order differential
equation with constant coefficients is written as
ay ′′ + by ′ + cy = 0 (16.46)
or
Ly = 0 (16.47)
We will denote a solution to the homogeneous equations as y H (t) to distinguish
it from a solution of
ay ′′ + by ′ + cy = f(t) (16.48)
If there are multiple solutions to the homogeneous equation we will number
them y H1 , y H2 , .... We will call any solution of (16.48) a particular
solution and denote it as y P (t). If there are multiple particular solutions
we will also number them if we need to.
Theorem 16.7. If y H (t) is a solution to
and y P (t) is a solution to
then
is also a solution to (16.50).
ay ′′ + by ′ + cy = 0 (16.49)
ay ′′ + by ′ + cy = f(t) (16.50)
y = y H (t) + y P (t) (16.51)