Math 3321 – Lecture 9 notes
Second order linear ODE
y′′ + p(x)y′ + q(x)y = f (x) (N )
y′′ + p(x)y′ + q(x)y = 0
(H )
The function f (x) is called the forcing function or the nonhomogeneous term.
The initial value problem:
y′′ + p(x)y′ + q(x)y = f (x)
y(a) = c
y′(a) = d
has a unique solution.
The zero function y(x) = 0 for all x ∈I (y ≡ 0) is a solution of (H). The zero function is
called the trivial solution. Any other solution is a non-trivial solution.
Any constant multiple of a solution of (H) is also a solution of (H).
The sum of any two solutions of (H) is also a solution of (H).
Definition: Let y = y1 (x) and y = y2 (x) be solutions of a second order linear
homogeneous differential equation. The function W defined by
,
=
=
−
is called the Wronskian of y1 and y2 .
Note: y1 and y2 are linearly independent iff W [y1 , y2 ] ≠ 0
− p ( x )dx
W (x) = Ce ∫
Definition: A pair of solutions y = y1 (x) and y = y2 (x) of equation (H) forms a
fundamental set of solutions if W [y1 , y2 ] ≠ 0 for all x ∈I . A fundamental set of
solutions is also called a solution basis.
Examples:
−4 x
1. y = 5xe is a solution of a second order homogeneous equation with constant
coefficients.
a) What is the equation?
b) What is the general solution?
−2 x
2. y = 4e sin(3x) is a solution of a second order homogeneous equation with
constant coefficients.
a) What is the equation?
b) What is the general solution?
y′′ + p(x)y′ + q(x)y = f (x) (N )
y′′ + p(x)y′ + q(x)y = 0
(H )
(H) is called the reduced equation of (N).
Theorem 1 in section 3.4 says that the difference of any two solutions of the
nonhomogeneous
equation (N) is a solution of its reduced equation (H).
Let y = y1 (x) and y = y2 (x) be linearly independent solutions of the reduced equation
(H) and let z = z(x) be a particular solution of (N). Then y(x) = C1 y1 (x) + C2 y2 (x) + z(x)
is a general solution of (N).
y(x) = C1 y1 (x) + C2 y2 (x) + z(x) consists of two parts:
Thm (Superposition Principle): If z = z f (x) and z = zg (x) are particular solutions of
y′′ + p(x)y′ + q(x)y = f (x)
and
y′′ + p(x)y′ + q(x)y = g(x)
respectively, then z(x) = z f (x) + zg (x) is a particular solution of
y′′ + p(x)y′ + q(x)y = f (x) + g(x)
Method of variation of parameters:
In order to find the general solution of (N), we need to find two things:
1. a linearly independent pair of solutions y1 and y2 of (H) and
2. a particular solution z of (N)
y2
y1
u′
=
−
f
and
v′
=
f
z is of the form z = uy1 + vy2 where
W
W
Examples:
1.
{ y (x) = x , y (x) = x } is a fundamental set of solutions of the reduced equation
y′′ −
2
1
4
2
5
8
y′ + 2 y = 4x 3 . Find a particular solution z of the equation.
x
x
2. { y1 (x) = e2 x , y2 (x) = e−3x } is a fundamental set of solutions of the reduced equation
y′′ + y′ − 6y = 3e2 x . Find the general solution of the equation.