To find the particular solution of inhomogeneous 2nd order linear ODEs with constant coefficients,
u" + au' + bu = g(x),
where g(x) is either a polynomial, exponential, or sinusoid (or a combination of these).

1.  Look at the Characteristic equation of homogeneous ODE
D(m) = m² + am + b = 0,
and find its roots m₁ and m₂.

2.  Identify which case is relevant from the tables below.

Case 1:  g(x) = a₀ + a₁ x + ... + a_n xⁿ
 

roots of D(m) = 0	Form of up(x)
(a) m1, m2 both nonzero	up(x) = A0 + A1 x + ... + An xn
(b)m1 = 0, m2 nonzero	up(x) = x(A0 + A1 x + ... + An xn)
(c) m1 = m2 = 0	up(x) = x2(A0 + A1 x + ... + An xn)

 

Case 2:  g(x) =  a e^kx
 

roots of D(m) = 0	Form of up(x)
(a) m1, m2 both not equal to k	up(x) = A ekx
(b) m1 = k, m2 not equal to k	up(x) = A x ekx
(c) m1 = m2 = k	up(x) = A x2 ekx

 
Case 3:  g(x) =  a cos(kx) + b sin(kx)
 

roots of D(m) = 0	Form of up(x)
(a) m1, m2 not equal to +/- ik	up(x) = A cos(kx) + B sin(kx)
(b) m1 = ik, m2 = -ik (complex conjugate roots)	up(x) = A x cos(kx) + B x sin(kx)

3.  Appy u_p(x) to the nonhomogeneous differential equation to determine coefficients.

4.  If g(x) = g₁(x) + g₂(x) + g₃(x) contains a combination of cases, then superpose solutions from each part.  I.e., solve u_p1(x) for nonhomogenous function g₁(x), u_p2(x) for nonhomogenous function g₂(x), and u_p3(x) for nonhomogenous function g₃(x), and the total particular solution is u_p(x) = u_p1(x) + u_p2(x) + u_p3(x),