Standard Form of nth Order LODE:
 

                               (1)

where the a_i’s are constant coefficients such that a_n is not zero, and f(t) is the forcing function. If f(t)=0 then equation (1) is said to be homogenous.
 

Solution:

The complete solution to equation (1) is a sum of two parts: the complementary solution x_c, and the particular solution x_p,
 

The complementary part, x_c (t) is the solution to the equation when f(t) is replaced by 0, i.e. made homogenous. It contains n arbitrary constants, which are evaluated later to fit boundary or initial conditions.

The particular part, x_p (t), is a solution which solves the original equation, i.e. when f(t) is put back into the equation. It does not have any arbitrary constant and must be linearly independent from the complementary solution.

The procedure involves the following steps:

1. Replace f(t) by zero and solve for x_c (t)
2. If f(t) is zero to begin with, then x_p (t)=0. Otherwise, find the particular solution that satisfies the original differential equation but is linearly independent from x_c (t).
3. Evaluate the arbitrary constants to satisfy the boundary conditions.

Complementary Solution:

First, assume that the complementary solution has the form:
 

where s is a parameter yet to be determined. Then the kth derivative of x_c (t) is given by
 

                                                         (4)

Thus after substitution of (3) into (1), while setting f(t)=0,
 

                                                      (5)

here p(s) is a polynomial function of s of degree n given by
 

                                                               (6)

Remarks:

1. p(s) is called the characteristic polynomial.
2. p(s)=0 is called the characteristic equation.
3. The roots of p(s)=0 are called the characteristic roots or eigenvalues

Let the roots of p(s) be {s₁, s₂, … , s_n}, then the complementary solution is going to be a sum of terms of exp(s_kt) multiplied by arbitrary coefficients and/or appropriate powers of t depending on how many times the root is repeated.

For each real root s_A that may have been repeated k_A times in the set, include the following terms to the complementary solution,

where A₀, …, A_kA are arbitrary coefficients.

For each pair of complex roots s_B= a + ib and s_C = a - ib that may have been repeated k_B=k_C times in the set, include the following terms to the complementary solution,

where B₀ , …, B_kB , C₀ , …, C_kC are arbitrary coefficients.
 
 

Particular Solution (using the method of undertermined coefficients)

Depending on the form of f(t), assume a similar form for x_p (t) but containing unknown coefficients. (If a term in the assumed x_p (t) matches any term of x_c (t), multiply these terms by t ^m where m is large enough to keep it independent x_c (t).

These assumed terms are then substituted into the differential equation. The following steps are then implemented:

1. The resulting algebraic equation is rearranged by collecting coefficients.
2. Coefficients from both sides of the equation are matched.
3. The unknown parameters are then evaluated by solving the resulting simultaneous equations.

Click here for an example.

     Tomas B. Co
     Associate Professor
     Department of Chemical Engineering
     Michigan Technological University
     1400 Townsend Avenue
     Houghton, MI 49931-1295

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