University
Maths Notes: Calculus – Variation of Parameters Method – Solving
Nonhomogeneous Differential Equations
We start from the nonhomogeneous differential
equation
The
associated homogeneous equation
has
fundamental – linearly independent – solutions
and
and
then the general solution of the associated homogeneous equation
is
where
and
are
constants. The general solution of the original nonhomogeneous
equation is
where
is
a particular solution of the original nonhomogeneous equation. The
method of variation of parameters looks for a particular solution
of
the form
which
means finding the functions
and
By
substituting
into
the original nonhomogeneous equation we obtain the simultaneous
equations
Solving these equations simultaneously gives
and
where
is
the determinant of the matrix
– this determinant is called the Wronskian.
Then
and
Summary
Find two fundamental solutions of the homogeneous
equation
Write down the form of the particular solution
Find
and
Write down the answer