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 y1(x), y2(x) of this equation. Obviously, the particular solutions depend on the coefficients of the differential equation. The Liouville formula establishesa connection between the Wronskian W(x), constructed on the basis of particular solutions y1(x), y2(x), and the coefficient ofa1(x) in the differential equation. Let W(x) be the Wronskian of the solutions y1(x), y2(x) of a linear second order homogeneous differential equation
in
which the functions a1(x) and a2(x) are
continuous on the interval [a,b].
Let the point x0 belong
to the interval [a,b].
Then for all
the Liouville
formula
is valid.
17. 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 y1(x), y2(x) is
a fundamental system of solutions, then the general
solution of
the second order equation is represented as
,
where C1, C2 are
arbitrary constants.
Note
that for a given fundamental system of solutions y1(x), y2(x),
we can construct the corresponding homogeneous differential equation.
For the case of a second order equation, it is expressed in terms of
the determinant:
= 0
Structure of the General Solution
The non-homogeneous differential equation of this type has the form
,
where p, q are constant numbers (that can be both as real as complex numbers). For each equation we can write the related homogeneous or complementary equation:
Theorem: The general solution of a non-homogeneous equation is the sum the general solution y0(x) of the related homogeneous equation and a particular solution y1(x) of the non-homogeneous equation:
Example:
Find the general solution of the equation y'' + y' −6y = 36x. Solution.
We will use the method of undetermined coefficients. The right side of the given equation is a linear function f(x) = ax + b. Therefore, we will look for a particular solution in the form
Then the derivatives are
Substituting this in the differential equation gives:
The last equation must be valid for all values of x, so the coefficients with the like powers of x in the right and left sides must be identical:
We find from this system that A = −6, B = −1. As a result, the particular solution is written as
Now we find the general solution of the homogeneous differential equation. Calculate the roots of the auxiliary characteristic equation:
Hence, the general solution of the related homogeneous equation is given by
Thus, the general solution of the initial nonhomogeneous equation is expressed by the formula
