Linear Systems of Differential Equations with Variable Coefficients

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A normal linear system of differential equations with variable coefficients can be written as

\[\frac{{d{x_i}}}{{dt}} = {x'_i} = \sum\limits_{j = 1}^n {{a_{ij}}\left( t \right){x_j}\left( t \right)} + {f_i}\left( t \right),\;\; i = 1,2, \ldots ,n, \]

where \({{x_i}\left( t \right)}\) are unknown functions, which are continuous and differentiable on an interval \(\left[ {a,b} \right].\) The coefficients \({{a_{ij}}\left( t \right)}\) and the free terms \({f_i}\left( t \right)\) are continuous functions on the interval \(\left[ {a,b} \right].\)

Using vector-matrix notation, this system of equations can be written as

\[{\mathbf{X'}}\left( t \right) = A\left( t \right){\mathbf{X}}\left( t \right) + {\mathbf{f}}\left( t \right),\]

where

\[{\mathbf{X}}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)}\\ \vdots \\ {{x_n}\left( t \right)} \end{array}} \right],\;\; A\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{a_{11}}\left( t \right)}&{{a_{12}}\left( t \right)}& \vdots &{{a_{1n}}\left( t \right)}\\ {{a_{21}}\left( t \right)}&{{a_{22}}\left( t \right)}& \vdots &{{a_{2n}}\left( t \right)}\\ \cdots & \cdots & \cdots & \cdots \\ {{a_{n1}}\left( t \right)}&{{a_{n2}}\left( t \right)}& \vdots &{{a_{nn}}\left( t \right)} \end{array}} \right],\;\; {\mathbf{f}}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( t \right)}\\ {{f_2}\left( t \right)}\\ \vdots \\ {{f_n}\left( t \right)} \end{array}} \right].\]

In the general case, the matrix \(A\left( t \right)\) and the vector functions \({\mathbf{X}}\left( t \right),\) \({\mathbf{f}}\left( t \right)\) can take both real and complex values.

The corresponding homogeneous system with variable coefficients in vector form is given by

\[{\mathbf{X'}}\left( t \right) = A\left( t \right){\mathbf{X}}\left( t \right).\]

Fundamental System of Solutions and Fundamental Matrix

The vector functions \({\mathbf{x}_1}\left( t \right),{\mathbf{x}_2}\left( t \right), \ldots ,{\mathbf{x}_n}\left( t \right)\) are linearly dependent on the interval \(\left[ {a,b} \right],\) if there are numbers \({c_1},{c_2}, \ldots ,{c_n},\) not all zero, such that the following identity holds:

\[{c_1}{\mathbf{x}_1}\left( t \right) + {c_2}{\mathbf{x}_2}\left( t \right) + \cdots + {c_n}{\mathbf{x}_n}\left( t \right) \equiv 0,\;\; \forall t \in \left[ {a,b} \right].\]

If this identity is satisfied only if

\[{c_1} = {c_2} = \cdots = {c_n} = 0,\]

the vector functions \({\mathbf{x}_i}\left( t \right)\) are called linearly independent on the given interval.

Any system of \(n\) linearly independent solutions \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right)\) is called a fundamental system of solutions.

A square matrix \(\Phi\left( t \right)\) whose columns are formed by linearly independent solutions \({\mathbf{x}_1}\left( t \right),{\mathbf{x}_2}\left( t \right), \ldots ,{\mathbf{x}_n}\left( t \right),\) is called the fundamental matrix of the system of equations. It has the following form:

\[\Phi \left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x_{11}}\left( t \right)}&{{x_{12}}\left( t \right)}& \vdots &{{x_{1n}}\left( t \right)}\\ {{x_{21}}\left( t \right)}&{{x_{22}}\left( t \right)}& \vdots &{{x_{2n}}\left( t \right)}\\ \cdots & \cdots & \cdots & \cdots \\ {{x_{n1}}\left( t \right)}&{{x_{n2}}\left( t \right)}& \vdots &{{x_{nn}}\left( t \right)} \end{array}} \right],\]

where \({{x_{ij}}\left( t \right)}\) are the coordinates of the linearly independent vector solutions \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right).\)

Note that the fundamental matrix \(\Phi \left( t \right)\) is nonsingular, i.e. there always exists the inverse matrix \({\Phi ^{ - 1}}\left( t \right).\) Since the fundamental matrix has \(n\) linearly independent solutions, after its substitution into the homogeneous system we obtain the identity

\[\Phi'\left( t \right) \equiv A\left( t \right)\Phi \left( t \right).\]

We multiply this equation on the right by the inverse function \({\Phi ^{ - 1}}\left( t \right):\)

\[\Phi'\left( t \right){\Phi ^{ - 1}}\left( t \right) \equiv A\left( t \right)\Phi \left( t \right){\Phi^{ - 1}}\left( t \right),\;\; \Rightarrow A\left( t \right) \equiv \Phi'\left( t \right){\Phi^{ - 1}}\left( t \right).\]

The resulting relation uniquely defines a homogeneous system of equations, given the fundamental matrix.

The general solution of the homogeneous system is expressed in terms of the fundamental matrix in the form

\[{\mathbf{X}_0}\left( t \right) = \Phi \left( t \right)\mathbf{C},\]

where \(\mathbf{C}\) is an \(n\)-dimensional vector consisting of arbitrary numbers.

Let us mention an interesting special case of homogeneous systems. It turns out that if the product of the matrix \(A\left( t \right)\) and the integral of this matrix is commutative, that is

\[A\left( t \right) \cdot \int\limits_a^t {A\left( \tau \right)dt} = \int\limits_a^t {A\left( \tau \right)dt} \cdot A\left( t \right),\]

the fundamental matrix \(\Phi\left( t \right)\) for such a system of equations is given by

\[\Phi \left( t \right) = {e^{\,\int\limits_a^t {A\left( \tau \right)d\tau } }}.\]

Such property is satisfied for symmetric matrices and, in particular, for diagonal matrices.

Wronskian and Liouville's Formula

The determinant of the fundamental matrix \(\Phi\left( t \right)\) is called the Wronskian of the system of solutions \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right):\)

\[W\left( t \right) = W\left[ {{\mathbf{x}_1},{\mathbf{x}_2}, \ldots ,{\mathbf{x}_n}} \right] = \left| {\begin{array}{*{20}{c}} {{x_{11}}\left( t \right)}&{{x_{12}}\left( t \right)}& \vdots &{{x_{1n}}\left( t \right)}\\ {{x_{21}}\left( t \right)}&{{x_{22}}\left( t \right)}& \vdots &{{x_{2n}}\left( t \right)}\\ \cdots & \cdots & \cdots & \cdots \\ {{x_{n1}}\left( t \right)}&{{x_{n2}}\left( t \right)}& \vdots &{{x_{nn}}\left( t \right)} \end{array}} \right|.\]

The Wronskian is useful to check the linear independence of solutions. The following rules apply:

The solutions \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right)\) of the homogeneous system form a fundamental system if and only if the corresponding Wronskian is not zero at any point \(t\) of the interval \(\left[ {a,b} \right].\)
The solutions \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right)\) are linearly dependent on the interval \(\left[ {a,b} \right]\) if and only if the Wronskian is identically zero on this interval.

The Wronskian of the solutions \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right)\) is given by Liouville's formula:

\[W\left( t \right) = W\left( a \right){e^{\,\int\limits_a^t {\text{tr}\left( {A\left( \tau \right)} \right)d\tau } }},\]

where \({\text{tr}\left( {A\left( \tau \right)} \right)}\) is the trace of the matrix \({A\left( \tau \right)},\) i.e. the sum of all diagonal elements:

\[\text{tr}\left( {A\left( \tau \right)} \right) = {a_{11}}\left( \tau \right) + {a_{22}}\left( \tau \right) + \cdots + {a_{nn}}\left( \tau \right).\]

Liouville's formula can be used to construct the general solution of the homogeneous system if a particular solution is known.

Method of Variation of Constants (Lagrange Method)

Now we consider the nonhomogeneous system that can be written in vector-matrix form as

\[\mathbf{X'}\left( t \right) = A\left( t \right)\mathbf{X}\left( t \right) + \mathbf{f}\left( t \right).\]

The general solution of such a system is the sum of the general solution \({\mathbf{X}_0}\left( t \right)\) of the corresponding homogeneous system and a particular solution \({\mathbf{X}_1}\left( t \right)\) of the nonhomogeneous system, that is

\[\mathbf{X}\left( t \right) = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right) = \Phi \left( t \right)\mathbf{C} + {\mathbf{X}_1}\left( t \right),\]

where \(\Phi \left( t \right)\) is a fundamental matrix, \(\mathbf{C}\) is an arbitrary vector.

The most common method for solving the nonhomogeneous systems is the method of variation of constants (Lagrange method). With this method, instead of the constant vector \(\mathbf{C}\) we consider the vector \(\mathbf{C}\left( t \right)\) whose components are continuously differentiable functions of the independent variable \(t,\) that is we assume

\[\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right).\]

Substituting this into the nonhomogeneous system, we find the unknown vector \(\mathbf{C}\left( t \right):\)

\[\mathbf{X'}\left( t \right) = A\left( t \right)\mathbf{X}\left( t \right) + \mathbf{f}\left( t \right),\;\; \Rightarrow \cancel{\Phi'\left( t \right)\mathbf{C}\left( t \right)} + \Phi \left( t \right)\mathbf{C'}\left( t \right) = \cancel{A\left( t \right)\Phi \left( t \right)\mathbf{C}\left( t \right)} + \mathbf{f}\left( t \right),\;\; \Rightarrow \Phi \left( t \right)\mathbf{C'}\left( t \right) = \mathbf{f}\left( t \right).\]

Given that the matrix \(\Phi \left( t \right)\) is nonsingular, we multiply this equation on the left by \({\Phi^{ - 1}}\left( t \right):\)

\[{\Phi^{ - 1}}\left( t \right)\Phi \left( t \right)\mathbf{C'}\left( t \right) = {\Phi^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C'}\left( t \right) = {\Phi^{ - 1}}\left( t \right)\mathbf{f}\left( t \right).\]

After integration we obtain the vector \(\mathbf{C}\left( t \right).\)