This section provides an alternative way how to find a solution of the inhomogeneous equation when a solution of a corresponding homogeneous equation is known.
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For instance,
construct a second-order linear ODE with variable coefficients that is arranged in
advance to be operationally factorable into the following pair of first-order equations
\begin{equation} \label{Eqfactor.1}
y_1 = L_1 \left[ x, \texttt{D} \right] y(x) = \left[ \texttt{D} + a(x) \right] y(x) ,
\end{equation}
and
\begin{equation} \label{Eqfactor.2}
L_2 \left[ x, \texttt{D} \right] y_1 (x) = \left[ x^n \texttt{D} + b(x) \right] y_1 (x) = f(x) ,
\end{equation}
where \( \displaystyle \texttt{D} = {\text d}/{\text d}x \) is the derivative operator, and 𝑎(x), b(x), and f(x) are given continuous functions.
Replacing y_{1}(x) from Eq.\eqref{Eqfactor.1}, we obtain
\[
\left[ x^n \texttt{D} + b(x) \right] \left[ \texttt{D} + a(x) \right] y(x) = f(x) .
\]
The operator multiplication of the term on the left-hand side requires
some care. For instance, the latter leads to the following second-order (linODE) with
variable coefficients:
\[
\left\{ x^n \texttt{D}^2 + \left[ b(x) + x^n a(x) \right] \texttt{D} + \left[ x^n \left( \texttt{D}\,a(x) \right) + a(x)\,b(x) \right] \right\} y(x) = f(x) .
\]
In the previous section, we factor the linear differential operators with variable coefficients of arbitrary order into a product of first order operator.
\( \texttt{D}^0 = \texttt{I} , \) the identity operator,
and corresponding n-th order inhomogeneous equations
\begin{equation} \label{EqIn.2}
L \left[ t, \texttt{D} \right] y = f \qquad \Longleftrightarrow \qquad y^{(n)} + a_{n-1}(t) \,y^{(n-1)} + \cdots + a_1 (t)\,y' + a_0 (t) \,y = f(t) .
\end{equation}
It is well known that no general formula exists for the solution of the variable coefficient equation \eqref{EqIn.2}.
If a fundamental set of solutions
to the homogeneous equation \( L \left[ t, \texttt{D} \right] y = 0 \) is known, then the variation of parameters provides an algorithm to determine aparticular solution to the inhomogeneous equation \eqref{EqIn.2}.
This section gives an alternative approach for its determination based on factorization of the given differential ooperator \( L \left[ t, \texttt{D} \right] \) into a product of first order differential operators:
where each \( L_i \left[ t, \texttt{D} \right] , \quad i=1,2,\ldots , n, \) is a first order differential operator. Note that such representation \eqref{EqIn.3} is not unique unless L is a constant coefficient operator---it is not our concern because we discusses this case in the next section. Moreover, the product of linear operators in Eq.\eqref{EqIn.3} is not commutative, so you have to preserve the order of their multiplications.
To make our exposition clear, we consider first second order variable coefficient equations, then the Third order equations, and finally, the general case.
Second order differential operators
A second order variable coefficient differential equation in normal form
\[
y'' + a_1 (t) \,y' + a_0 (t)\,y = f(t)
\]
can be easily breaken into the system of first order equations
The latter equation is a Riccati equation in p(t), so no general formula of the solution is known. However, when one solution φ_{1}(t) of the homogeneous equation
\( y'' + a_1 y' + a_0 y = 0 \) is known, then the Riccati equation for p(t) is solvable, and we obtain an explicit formula for a particular solution
Let
\( \varphi_1 , \varphi_2 , \ldots , \varphi_n \) be the fundamental set of solutions to the n-th order linear variable coefficient homogeneous equation \( L \left[ t, \texttt{D} \right] y = 0 , \) where the operator L is defined by Eq.\eqrefl{EqIn.1}. For any k functions from this fundamental set of solutions, we define the determinant:
Mejlbro, L., Solution of linear ordinary differential equations by means of the method of variation of arbitrary constants, International Journal of Mathematical Education in Science and Technology, 1997, Vol. 28, No. 3, pp. 321--331. doi: 10.1080/0020739970280302
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