Standard Form of nth Order LODE:
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:
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
Thus after substitution of (3) into (1), while setting f(t)=0,
here p(s) is a polynomial function of s of degree
n
given by
Remarks:
Let the roots of p(s) be {s_{1}, s_{2},
… , s_{n}}, then the complementary solution is going to be
a sum of terms of exp(s_{k}t) 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 A_{0}, …, 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 B_{0 }, …, B_{kB },_{ }C_{0 },
…, 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:








A t^{ m} exp( r t ) sin ( q t ) 
(E_{0} + E_{1} t + … + E_{m} t^{m}) exp( r t ) sin ( q t ) 
Click here for an example.
Tomas B. Co
Associate Professor
Department of Chemical Engineering
Michigan Technological University
1400 Townsend Avenue
Houghton, MI 499311295