A
Level Maths Notes: FP1 – Solving Second Order Linear Non -
Homogeneous Differential Equations
Any differential equation of the form
(1)
is a second order differential equations and there is a standard
technique for solving any equation of this sort. We solve the
homogeneous equation
(2)
first for the 'complementary' solution
We
assume a solution of the form
and
substitute this into (2). We extract the non zero factor – since no
exponential is zero for any finite x -
to
obtain a quadratic equation. We solve this equation to obtain
solutions
and
and
then the general solution is
Having
found the complementary solution
we
now find a particular solution
to
(1) by assuming a solution 'similar' to
then
the general solution to (1) is given by
Example: Solve the equation
(2)
subject to
and
at
Substitution of the above expressions into
gives
We can factor out the nonzeroto
obtain
Becauseis
non zero we can divide by it to obtain
and
we factorise this expression to obtain
and
solve to obtain
and
The
complementary solution is then
To find a particular solution we assume
since
is
a polynomial of degree one.
and
Substitute
these into (1)
Equating coefficients of
gives
Equating constants gives
when
implies
(3)
atimplies
(4)
(3)+(4) gives
then
from (3)
The solution is
