Differential Equations

Systems of Equations

Linear Systems of Differential Equations with Variable Coefficients

Page 1
Theory
Page 2
Problems 1-3

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

\[
{\frac{{d{x_i}}}{{dt}} = {x’_i} }={ \sum\limits_{j = 1}^n {{a_{ij}}\left( t \right){x_j}\left( t \right)} + {f_i}\left( t \right),\;\;}\kern-0.3pt
{i = 1,2, \ldots ,n,}
\]

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

\[
{{\mathbf{X}}\left( t \right) = \left[ {\begin{array}{*{20}{c}}
{{x_1}\left( t \right)}\\
{{x_2}\left( t \right)}\\
\vdots \\
{{x_n}\left( t \right)}
\end{array}} \right],\;\;}\kern-0.3pt
{{A\left( t \right) \text{ = }}\kern0pt{\left[ {\begin{array}{*{20}{c}}
{{a_{11}}\left( t \right)}&{{a_{12}}\left( t \right)}& \vdots &{{a_{1n}}\left( t \right)}\\
{{a_{21}}\left( t \right)}&{{a_{22}}\left( t \right)}& \vdots &{{a_{2n}}\left( t \right)}\\
\cdots & \cdots & \cdots & \cdots \\
{{a_{n1}}\left( t \right)}&{{a_{n2}}\left( t \right)}& \vdots &{{a_{nn}}\left( t \right)}
\end{array}} \right],\;\;}}\kern-0.3pt
{{\mathbf{f}}\left( t \right) = \left[ {\begin{array}{*{20}{c}}
{{f_1}\left( t \right)}\\
{{f_2}\left( t \right)}\\
\vdots \\
{{f_n}\left( t \right)}
\end{array}} \right].}
\]

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:

\[
{{c_1}{\mathbf{x}_1}\left( t \right) + {c_2}{\mathbf{x}_2}\left( t \right) + \cdots }+{ {c_n}{\mathbf{x}_n}\left( t \right) \equiv 0,\;\;}\kern-0.3pt
{\forall t \in \left[ {a,b} \right].}
\]

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:

\[{\Phi \left( t \right) \text{ = }}\kern0pt{ \left[ {\begin{array}{*{20}{c}}
{{x_{11}}\left( t \right)}&{{x_{12}}\left( t \right)}& \vdots &{{x_{1n}}\left( t \right)}\\
{{x_{21}}\left( t \right)}&{{x_{22}}\left( t \right)}& \vdots &{{x_{2n}}\left( t \right)}\\
\cdots & \cdots & \cdots & \cdots \\
{{x_{n1}}\left( t \right)}&{{x_{n2}}\left( t \right)}& \vdots &{{x_{nn}}\left( t \right)}
\end{array}} \right],}\]

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, i.e.

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

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 in the case of symmetric matrices and, in particular, in the case of 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 \({\mathbf{x}_1}\left( t \right),\) \({\mathbf{x}_2}\left( t \right), \ldots ,\) \({\mathbf{x}_n}\left( t \right):\)

\[
{W\left( t \right) }={ W\left[ {{\mathbf{x}_1},{\mathbf{x}_2}, \ldots ,{\mathbf{x}_n}} \right] \text{ = }}\kern0pt
{\left| {\begin{array}{*{20}{c}}
{{x_{11}}\left( t \right)}&{{x_{12}}\left( t \right)}& \vdots &{{x_{1n}}\left( t \right)}\\
{{x_{21}}\left( t \right)}&{{x_{22}}\left( t \right)}& \vdots &{{x_{2n}}\left( t \right)}\\
\cdots & \cdots & \cdots & \cdots \\
{{x_{n1}}\left( t \right)}&{{x_{n2}}\left( t \right)}& \vdots &{{x_{nn}}\left( t \right)}
\end{array}} \right|.}
\]

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:

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

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 systems that can be written in the 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, i.e.

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

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 \(t,\) i.e. 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 \(\mathbf{C}\left( t \right):\)

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

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 \(\mathbf{C}\left( t \right).\)

Solved Problems

Click on problem description to see solution.

 Example 1

Write the linear system of equations with the following solutions:

\[
{{\mathbf{x}_1}\left( t \right) = \left[ {\begin{array}{*{20}{c}}
2\\
t
\end{array}} \right],\;\;}\kern-0.3pt
{{\mathbf{x}_2}\left( t \right) = \left[ {\begin{array}{*{20}{c}}
t\\
{{t^2}}
\end{array}} \right],\;\;}\kern-0.3pt
{t \ne 0.}
\]

 Example 2

Find a fundamental matrix of the system of differential equations

\[{\frac{{dx}}{{dt}} = x + ty,\;\;}\kern-0.3pt{\frac{{dy}}{{dt}} = tx + y.}\]

making sure that the coefficient matrix \(A\left( t \right)\) commutes with its integral.

 Example 3

Find the general solution of the system

\[
{\frac{{dx}}{{dt}} = – tx + y,\;\;}\kern-0.3pt
{\frac{{dy}}{{dt}} = \left( {1 – {t^2}} \right)x + ty,\;\;}\kern-0.3pt
{x \gt 0,}
\]

if one solution is known:

\[{\mathbf{X}_1}\left( t \right) = \left[ {\begin{array}{*{20}{c}}
{{x_1}\left( t \right)}\\
{{y_1}\left( t \right)}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1\\
t
\end{array}} \right].\]
Page 1
Theory
Page 2
Problems 1-3