A Level Maths Notes: FP1 – Solving Second Order Linear Homogeneous Differential Equations
Any differential equation of the form
is
a second order differential equations and there is a standard
technique for solving any equation of this sort. We assume a solution
of the form
and
substitute this into the equation. We extract the non zero factor –
since no exponential is zero for any finite x -
to
obtain a quadratic equation. We solve this equation obtain
solutions
and
and
then the general solution is
and
may
be evaluated given suitable boundary conditions, for example
Example: Solve the equation
(1)
subject to
and
at
Substitution of the above expressions into (1) 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
general solution is then
We
now have to find
when
implies
(2)
atimplies
(3)
(2)+(3) gives
then
from (2)
The solution is
Example: Solve the equation
(1)
subject towhenand
at
Substitution of the above expressions into (1) gives
We can move
to
the left hand side and 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
general solution is then
We
now have to find
whenimplies
(2)
at
implies
(3)
(2)-(3)
gives
then
from (2)
The solution is
