Second Order Linear Homogeneous Differential Equations with Variable Coefficients
A linear homogeneous second order equation with variable coefficients can be written as
where
Linear Independence of Functions. Wronskian
The functions
holds. If this identity is satisfied only when
For the case of two functions, the linear independence criterion can be written in a simpler form: The functions
Otherwise, when
Let
is called the Wronski determinant or Wronskian for this system of functions.
Wronskian Test.
If the system of functions
It follows from here that if the Wronskian is nonzero at least at one point in the interval
Fundamental System of Solutions
A set of two linearly independent particular solutions of a linear homogeneous second order differential equation forms its fundamental system of solutions.
If
where
Note that for a given fundamental system of solutions
Liouville's Formula
Thus, as noted above, the general solution of a homogeneous second order differential equation is a linear combination of two linearly independent particular solutions
Obviously, the particular solutions depend on the coefficients of the differential equation. The Liouville formula establishes a connection between the Wronskian
Let
in which the functions
is valid.
Practical Methods for Solving Second Order Homogeneous Equations with Variable Coefficients
Unfortunately, there is no general method for finding a particular solution. Usually this is done by guessing.
If a particular solution
Another way to reduce the order is based on the Liouville formula. In this case, a particular solution
Solved Problems
Click or tap a problem to see the solution.
Example 1
Investigate whether the functions
Example 2
Find the Wronskian of the system of functions
Example 1.
Investigate whether the functions
Solution.
We form the quotient of two functions:
It is seen that this ratio is not equal to a constant, but depends on
Example 2.
Find the Wronskian of the system of functions
Solution.
The Wronskian of the system of two functions is calculated by the formula:
Substituting the given functions and their derivatives, we obtain
It follows from here, that functions