Lecture Notes in Differential Equations - Bruce E. Shapiro

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3. If f(t) = e rt and r is a root of the characteristic equation, but is not
a repeated root, try
y P = Ate rt (19.16)
4. If f(t) = e rt and r is a repeated root of the characteristic equation,
try
y P = At 2 e rt (19.17)
5. If f(t) = α sin ωt + β cos ωt, where α, β, ω ∈ R, and neither sin ωt nor
cos ωt are solutions of the homogeneous equation, try
y P = A cos ωt + B sin ωt (19.18)
If (19.18) is a solution of the homogeneous equation, instead try
y P = t(A cos ωt + B sin ωt) (19.19)
6. If f is a product of polynomials, exponentials, and/or sines and
cosines, use a product of polynomials, exponentials, and/or sines and
cosines. If any of the terms in the product is a solution of the homogeneous
equation, multiply the entire solution by t or t 2 , whichever
ensures that no terms in the guess are a solution of Ly = 0.
Example 19.2. Solve
The characteristic equation is
hence
y ′′ + y ′ − 6y = 2t (19.20)
r 2 + r − 6 = (r − 2)(r + 3) = 0 (19.21)
y H = C 1 e 2t + C 2 e −3t (19.22)
Since the forcing function (right-hand side of the equation) is 2t we try a
particular function of
y P = At + B (19.23)
Differentiating,
Substituting back into the differential equation,
y P ′ = A (19.24)
y P ′′ = 0 (19.25)
0 + A − 6(At + B) = 6t (19.26)
−6At + A + B = 6t (19.27)