Lecture Notes in Differential Equations - Bruce E. Shapiro

196 LESSON 22. NON-HOMOG. EQS.W/ CONST. COEFF.
and the particular solution, in either case, is
∫ (∫
)
y P = e r2t e (r1−r2)t q(t)e −r1t dt dt (22.90)
Returning to the original ODE (22.67), before we defined q(t) = f(t)/a,
y P = 1 ∫ (∫
)
a er2t e (r1−r2)t f(t)e −r1t dt dt (22.91)
We have just proven the following two theorems.
Theorem 22.6. The general solution of the second order homogeneous
linear differential equation with constant coefficients,
ay ′′ + by ′ + cy = 0 (22.92)
is
y H =
{
C 1 e r1t + C 2 e r2t if r 1 ≠ r 2
(C 1 t + C 2 )e rt if r 1 = r 2 = r
where r 1 and r 2 are the roots of the characteristic equation
(22.93)
ar 2 + br + c = 0 (22.94)
Theorem 22.7. A particular solution for the second order linear differential
equation with constant coefficients
ay ′′ + by ′ + cy = f(t) (22.95)
is
y P = 1 a er2t ∫
e (r1−r2)t (∫
)
f(t)e −r1t dt dt (22.96)
where r 1 and r 2 are the roots of the characteristic equation
ar 2 + br + c = 0 (22.97)
Example 22.2. Solve the initial value problem
⎫
y ′′ − 9y = e 2t ⎪⎬
y(0) = 1
⎪⎭
y ′ (0) = 2
(22.98)