Lecture Notes in Differential Equations - Bruce E. Shapiro

194 LESSON 22. NON-HOMOG. EQS.W/ CONST. COEFF.
Derivation of General Solution. We are now ready to find a general
formula for the solution of
ay ′′ + by ′ + cy = f(t) (22.67)
where a ≠ 0, for any function f(t). We begin by dividing by a,
y ′′ + By ′ + Cy = q(t) (22.68)
where B = b/a, C = c/a, and q(t) = f(t)/a. By the factorization theorem
(theorem (22.4)), (22.68) is equivalent to
(D − r 1 )(D − r 2 )y = q(t) (22.69)
where
r 1,2 = 1 2
(
−B ± √ )
B 2 − 4C
(22.70)
Defining
z = (D − r 2 )y = y ′ − r 2 y (22.71)
equation (22.69) becomes
(D − r 1 )z = q(t) (22.72)
or equivalently,
z ′ − r 1 z = q(t) (22.73)
This is a first order linear ODE in z(t) with integrating factor
(∫ )
µ(t) = exp −r 1 dt = e −r1t (22.74)
and the solution of (22.73) is
z = 1
µ(t)
Using (22.71) this becomes
(∫
q(t)µ(t)dt + C 1
)
(22.75)
y − r 2 y = p(t) (22.76)
where
p(t) = 1 (∫
µ(t)
q(t)µ(t)dt + C 1
)
Equation (22.76) is a first order linear ODE in y(t); its solution is
y(t) = 1 (∫
)
p(t)ν(t)dt + C 2
ν(t)
(22.77)
(22.78)