# 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:

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

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 $$\mathbf{C}\left( t \right):$$

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

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 or tap 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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