# Linear Nonhomogeneous Systems of Differential Equations with Constant Coefficients

## Trigonometry # Linear Nonhomogeneous Systems of Differential Equations with Constant Coefficients

A normal linear inhomogeneous system of $$n$$ equations with constant coefficients can be written as

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

where $$t$$ is the independent variable (often $$t$$ is time), $${{x_i}\left( t \right)}$$ are unknown functions which are continuous and differentiable on an interval $$\left[ {a,b} \right]$$ of the real number axis $$t,$$ $${a_{ij}}\left( {i,j = 1, \ldots ,n} \right)$$ are the constant coefficients, $${f_i}\left( t \right)$$ are given functions of the independent variable $$t.$$ We assume that the functions $${{x_i}\left( t \right)},$$ $${{f_i}\left( t \right)}$$ and the coefficients $${a_{ij}}$$ may take both real and complex values.

We introduce the following vectors:

$\mathbf{X}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)}\\ \vdots \\ {{x_n}\left( t \right)} \end{array}} \right],\;\; \mathbf{f}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( t \right)}\\ {{f_2}\left( t \right)}\\ \vdots \\ {{f_n}\left( t \right)} \end{array}} \right]$

and the square matrix

$A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \vdots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \vdots &{{a_{2n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{a_{n1}}}&{{a_{n2}}}& \vdots &{{a_{nn}}} \end{array}} \right].$

Then the system of equations can be written in a more compact matrix form as

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

For nonhomogeneous linear systems, as well as in the case of a linear homogeneous equation, the following important theorem is valid:

The general solution $$\mathbf{X}\left( t \right)$$ of the nonhomogeneous system is the sum of the general solution $${\mathbf{X}_0}\left( t \right)$$ of the associated homogeneous system and a particular solution $${\mathbf{X}_1}\left( t \right)$$ of the nonhomogeneous system:

$\mathbf{X}\left( t \right) = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right).$

Methods of solutions of the homogeneous systems are considered on other web-pages of this section. Therefore, below we focus primarily on how to find a particular solution.

Another important property of linear inhomogeneous systems is the principle of superposition, which is formulated as follows:

If $${\mathbf{X}_1}\left( t \right)$$ is a solution of the system with the inhomogeneous part $${\mathbf{f}_1}\left( t \right),$$ and $${\mathbf{X}_2}\left( t \right)$$ is a solution of the same system with the inhomogeneous part $${\mathbf{f}_2}\left( t \right),$$ then the vector function

$\mathbf{X}\left( t \right) = {\mathbf{X}_1}\left( t \right) + {\mathbf{X}_2}\left( t \right)$

is a solution of the system with the inhomogeneous part

$\mathbf{f}\left( t \right) = {\mathbf{f}_1}\left( t \right) + {\mathbf{f}_2}\left( t \right).$

The most common methods of solution of the nonhomogeneous systems are the method of elimination, the method of undetermined coefficients (in the case where the function $$\mathbf{f}\left( t \right)$$ is a vector quasi-polynomial), and the method of variation of parameters. Consider these methods in more detail.

## Elimination Method

This method allows to reduce the normal nonhomogeneous system of $$n$$ equations to a single equation of $$n$$th order. This method is useful for solving systems of order $$2.$$

## Method of Undetermined Coefficients

The method of undetermined coefficients is well suited for solving systems of equations, the inhomogeneous part of which is a quasi-polynomial.

A real vector quasi-polynomial is a vector function of the form

$\mathbf{f}\left( t \right) = {e^{\alpha t}}\left[ {\cos \left( {\beta t} \right){\mathbf{P}_m}\left( t \right) + \sin \left( {\beta t} \right){\mathbf{Q}_m}\left( t \right)} \right],$

where $$\alpha,$$ $$\beta$$ are given real numbers, and $${{\mathbf{P}_m}\left( t \right)},$$ $${{\mathbf{Q}_m}\left( t \right)}$$ are vector polynomials of degree $$m.$$ For example, a vector polynomial $${{\mathbf{P}_m}\left( t \right)}$$ is written as

${\mathbf{P}_m}\left( t \right) = {\mathbf{A}_0} + {\mathbf{A}_1}t + {\mathbf{A}_2}{t^2} + \cdots + {\mathbf{A}_m}{t^m},$

where $${\mathbf{A}_0},$$ $${\mathbf{A}_2}, \ldots ,$$ $${\mathbf{A}_m}$$ are $$n$$-dimensional vectors ($$n$$ is the number of equations in the system).

In the case when the inhomogeneous part $$\mathbf{f}\left( t \right)$$ is a vector quasi-polynomial, a particular solution is also given by a vector quasi-polynomial, similar in structure to $$\mathbf{f}\left( t \right).$$

For example, if the nonhomogeneous function is

$\mathbf{f}\left( t \right) = {e^{\alpha t}}{\mathbf{P}_m}\left( t \right),$

a particular solution should be sought in the form

${\mathbf{X}_1}\left( t \right) = {e^{\alpha t}}{\mathbf{P}_{m + k}}\left( t \right),$

where $$k = 0$$ in the non-resonance case, i.e. when the index $$\alpha$$ in the exponential function does not coincide with an eigenvalue $${\lambda _i}.$$ If the index $$\alpha$$ coincides with an eigenvalue $${\lambda _i},$$ i.e. in the so-called resonance case, the value of $$k$$ is chosen to be equal to the greatest length of the Jordan chain for the eigenvalue $${\lambda _i}.$$ In practice, $$k$$ can be taken as the algebraic multiplicity of $${\lambda _i}.$$

Similar rules for determining the degree of the polynomials are used for quasi-polynomials of kind

${e^{\alpha t}}\cos \left( {\beta t} \right),\;\; {e^{\alpha t}}\sin\left( {\beta t} \right).$

Here the resonance case occurs when the number $$\alpha + \beta i$$ coincides with a complex eigenvalue $${\lambda _i}$$ of the matrix $$A.$$

After the structure of a particular solution $${\mathbf{X}_1}\left( t \right)$$ is chosen, the unknown vector coefficients $${A_0},$$ $${A_1}, \ldots ,$$ $${A_m}, \ldots ,$$ $${A_{m + k}}$$ are found by substituting the expression for $${\mathbf{X}_1}\left( t \right)$$ in the original system and equating the coefficients of the terms with equal powers of $$t$$ on the left and right side of each equation.

## Method of Variation of Constants

The method of variation of constants (Lagrange method) is the common method of solution in the case of an arbitrary right-hand side $$\mathbf{f}\left( t \right).$$

Suppose that the general solution of the associated homogeneous system is found and represented as

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

where $$\Phi \left( t \right)$$ is a fundamental system of solutions, i.e. a matrix of size $$n \times n,$$ whose columns are formed by linearly independent solutions of the homogeneous system, and $$\mathbf{C} = {\left( {{C_1},{C_2}, \ldots ,{C_n}} \right)^T}$$ is the vector of arbitrary constant numbers $${C_i}\left( {i = 1, \ldots ,n} \right).$$

We replace the constants $${C_i}$$ with unknown functions $${C_i}\left( t \right)$$ and substitute the function $$\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right)$$ in the nonhomogeneous system of equations:

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

Since the Wronskian of the system is not equal to zero, then there exists the inverse matrix $${\Phi ^{ - 1}}\left( t \right).$$ Multiplying the last equation on the left by $${\Phi ^{ - 1}}\left( t \right),$$ we obtain:

${\Phi ^{ - 1}}\left( t \right)\Phi \left( t \right)\mathbf{C'}\left( t \right) = {\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C'}\left( t \right) = {\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C}\left( t \right) = {\mathbf{C}_0} + \int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt} ,$

where $${\mathbf{C}_0}$$ is an arbitrary constant vector.

Then the general solution of the nonhomogeneous system can be written as

$\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right) = \Phi \left( t \right){\mathbf{C}_0} + \Phi \left( t \right)\int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt} = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right).$

We see that a particular solution of the nonhomogeneous equation is represented by the formula

${\mathbf{X}_1}\left( t \right) = \Phi \left( t \right)\int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt}.$

Thus, the solution of the nonhomogeneous equation can be expressed in quadratures for any inhomogeneous term $$\mathbf{f}\left( t \right).$$ In many problems, the corresponding integrals can be calculated analytically. This allows us to express the solution of the nonhomogeneous system explicitly.