Differential Equations

Systems of Equations

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Linear Systems of Differential Equations with Variable Coefficients

  • 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, 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 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, 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 \(\mathbf{C}\left( t \right):\)

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

    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 a problem to see the 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:
    \[{\mathbf{X}_1}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{y_1}\left( t \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ t \end{array}} \right].\]
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    Problems 1-3