Then the transformation reduces the equation into an order equation.
when .
If are linearly independent (LI) solutions of (2) and if for , then
is a basis of solutions of (1), where .
Proof
Let .
Suppose is a solution of (1), then
Sorting according to :
when .
This is an order Differential Equation (DE) and it has LI solutions, .
Since , .
Therefore are the solutions of (1).
It remains to show that the above solutions form a basis.
Suppose .
since are LI.
, and hence for all .
are LI and form a basis.
And if for , then
is a basis of solution of (1), where
Method of Variation of Parameters
Theorem
Let
where , , and are continuous functions of defined on an interval . If are two linearly independent solutions corresponding to the homogeneous equation,
then, a particular solution of (1) is given by,
where
where is the Wronskian of .
This is an order differential equation and it has linearly independent solutions namely .
Since .
Therefore, are solutions of (1). It remains to show that the above solution is a basis.
Suppose
since
(differentiating)
for since are linearly independent.
and hence for all .
are linearly independent and hence a basis.
Method of Variation of Parameters
Theorem
Set where , , and are continuous functions of defined on an interval . If are two linearly independent solutions corresponding to the homogeneous equation , then a particular solution of (1) is given by where