Lecture Notes in Differential Equations - Bruce E. Shapiro

414 APPENDIX C. SUMMARY OF METHODS
To find y P make an educated guess based on the form form of f(t). The
educated guess should be the product
y P = P (t)S(t)E(t)
where P (t) is a polynomial of the same order as in f(t). S(t) = r n (A sin rt+
B cos rt) is present only if there are trig functions in rt in f(t), and n is the
multiplicity of r as a root of the characteristic equation (n = 0 if r is not a
root). E(t) = r n e rt is present only if there is an exponential in rt in f(t).
If f(t) = f 1 (t) + f 2 (t) + · · · then solve each of the equations
ay ′′ + by ′ + cy = f i (t)
separately and add all of the particular solutions together to get the complete
particular solution.
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 r1t ∫
t
e r2−r1 sds + er1t
a
where r 1 and r 2 are the roots of ar 2 + br + c = 0.
∫
t
∫
e r2−r1 s e −r2u f(u)duds
s
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
(D − r 2 )y = z