# Differential Equations

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

Theory

Problems 1-6

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),\;\;}\kern-0.3pt

{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],\;\;}\kern0pt

{\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

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

\[{{e^{\alpha t}}\cos \left( {\beta t} \right),\;\;}\kern0pt{{e^{\alpha t}}\sin\left( {\beta t} \right).}\]
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:

\[\require{cancel}

{\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.

## Solved Problems

Click on problem description to see solution.

### ✓ Example 1

Solve the system of equations by elimination:

\[{x’ = x + 2y + {e^{ – 2t}},\;\;}\kern-0.3pt{y’ = 4x – y.}\]

### ✓ Example 2

Solve the system of equations by the method of undetermined coefficients:

\[{\frac{{dx}}{{dt}} = 2x + y,\;\;}\kern-0.3pt{\frac{{dy}}{{dt}} = 3y + t{e^t}.}\]

### ✓ Example 3

Solve the system of equations by the method of undetermined coefficients:

\[{\frac{{dx}}{{dt}} = x + {e^t},\;\;}\kern-0.3pt{\frac{{dy}}{{dt}} = x + y – {e^t}.}\]

### ✓ Example 4

Solve the system of equations by the method of undetermined coefficients:

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

### ✓ Example 5

Solve the system of equations by variation of parameters:

\[{\frac{{dx}}{{dt}} = y + \frac{1}{{\cos t}},\;\;}\kern-0.3pt{\frac{{dy}}{{dt}} = – x.}\]

### ✓ Example 6

Solve the linear nonhomogeneous system by variation of constants:

\[

{\frac{{dx}}{{dt}} = 2x – y + {e^{2t}},\;\;}\kern-0.3pt

{{\frac{{dy}}{{dt}} = 6x – 3y }+{ {e^t} + 1.}}

\]

Theory

Problems 1-6

### Related Pages

- Linear Homogeneous Systems of Differential Equations with Constant Coefficients
- Method of Eigenvalues and Eigenvectors
- Second Order Linear Homogeneous Differential Equations with Constant Coefficients
- Second Order Linear Nonhomogeneous Differential Equations with Constant Coefficients
- The Indefinite Integral and Basic Formulas of Integration. Table of Integrals.
- Properties of Matrices
- Properties of Determinants
- Complex Numbers
- Systems of Linear Equations