The Computable Differential Equation Lecture ... - Bruce E. Shapiro

196
CHAPTER 10.
APPENDIX ON ANALYTIC METHODS
(MATH 280 IN A NUTSHELL)
General Non-homogeneous Linear Equation with Constant Coefficients
To solve
ay ′′ + by ′ + cy = f(t)
where a, b, c are constants for a general function f(t), the solution is
∫
∫ ∫
y = Ae r1t + Be r 1t
e r 2−r 1
sds + er 1t
e r 2−r 1
s e −r2u f(u)duds
a
t
where r 1 and r 2 are the roots of ar 2 + br + c = 0.
An alternative method is to factor the equation into the form
and make the substitution
(D − r 1 )(D − r 2 )y = f(t)
z = (D − r 2 )y
This reduces the second order equation in y to a first order linear equation in z.
Solve the equation
(D − r 1 )z = f(t)
for z, then solve the equation
for y once z is known.
Method of Reduction of Order
(D − r 2 )y = z
If one solution y 1 is known for the differential equation
y ′′ + p(t)y ′ + q(t)y = 0
then a second solution is given by
∫ W (y1 , y 2 ))(t)
y 2 (t) = y 1 (t)
y 1 (t) 2 dt
where the Wronskian is given by Abel’s formula
( ∫
W (y 1 , y 2 )(t) = Cexp −
t
s
)
p(s)ds
Method of Variation of Parameters
To find a particular solution to
y ′′ + p(t)y ′ + q(t)y = r(t)
when a pair of linearly independent solutions to the homogeneous equation
are already known,
∫
y p = −y 1 (t)
t
y ′′ + p(t)y ′ + q(t)y = 0
∫
y 2 (s)r(s)
W (y 1 , y 2 )(s) ds + y y 1 (s)r(s)
2(t)
t W (y 1 , y 2 )(s) ds
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007