Linear Systems

students, imagine tracking how two things change together at the same time 🌱🚗. One could be the number of gallons of water in a tank, and the other could be the temperature of a room. In differential equations, a linear system is a group of differential equations where the unknown functions appear in a linear way. This lesson introduces the main ideas, shows how to read linear systems, and explains why they matter in the wider study of systems of differential equations.

What is a linear system?

A system of differential equations is a set of equations that uses more than one unknown function. For example, if the unknowns are $x(t)$ and $y(t)$, a system might look like this:

$\frac{dx}{dt} = 3x - 2y$

$\frac{dy}{dt} = 4x + y$

This is called a linear system because each equation is linear in the unknown functions $x(t)$ and $y(t)$. That means:

the functions $x$ and $y$ are not multiplied together,
the functions are not squared or put inside a sine, exponential, or other nonlinear function,
the derivatives appear only to the first power,
coefficients may depend on $t$, but the unknown functions themselves stay linear.

A more general form is

$\frac{dx}{dt}$ = a(t)x + b(t)y + f(t)

$\frac{dy}{dt}$ = c(t)x + d(t)y + g(t)

If $f(t)$ and $g(t)$ are both zero, the system is called homogeneous. If at least one of them is not zero, the system is nonhomogeneous.

The word “linear” is important because linear systems have special patterns and tools that make them easier to study than nonlinear systems. This is one reason they appear everywhere in science and engineering.

Why linear systems matter

students, many real-world situations involve several quantities affecting each other at the same time. Linear systems give a first model for these interactions 📈.

Here are a few examples:

Population models: two species may interact through competition or cooperation.
Electrical circuits: voltage and current may influence each other.
Mechanics: position and velocity often form a system.
Chemistry: the amounts of two substances can change together through reactions.

Even when a real situation is more complicated, a linear system can still be useful as an approximation near a steady state. That means linear systems often help describe what happens when a system is close to balance.

A big advantage of linear systems is that they obey the principle of superposition. This means that if one solution works and another solution works, then certain combinations of them also work. For homogeneous linear systems, this property is especially powerful.

Writing linear systems in matrix form

A linear system can be written more compactly using matrices. This helps organize the coefficients and makes the structure easier to see.

Suppose we have

$\frac{dx}{dt} = ax + by$

$\frac{dy}{dt} = cx + dy$

Let

$\mathbf{x}$(t) = $\begin{pmatrix}$ x(t) \ y(t) $\end{pmatrix}$

Then the system becomes

$\mathbf{x}' = A\mathbf{x}$

where

A = $\begin{pmatrix}$ a & b \ c & d $\end{pmatrix}$

This is the matrix form of a linear homogeneous system.

If the system has extra forcing terms, it can be written as

$\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t)$

where

$\mathbf{f}$(t) = $\begin{pmatrix}$ f(t) \ g(t) $\end{pmatrix}$

Using matrix form is not just a shortcut. It reveals a deeper connection between differential equations and linear algebra. That connection becomes very important when studying eigenvalues and eigenvectors later in the topic.

How to check whether a system is linear

To decide whether a system is linear, students, check the unknown functions and their derivatives carefully 🔎.

A system is linear if it fits this pattern:

each unknown function appears only to the first power,
unknowns are not multiplied together,
coefficients depend only on the independent variable, often $t$,
derivatives appear linearly as well.

Compare these examples:

Linear example

$\frac{dx}{dt}$ = 2x - 5y + $\sin$ t

$\frac{dy}{dt} = x + 3y$

This is linear because $x$ and $y$ appear only in first degree, and the coefficients depend on $t$ or are constants.

Nonlinear example

$\frac{dx}{dt} = x^2 - y$

$\frac{dy}{dt} = xy + 1$

This is not linear because of $x^2$ and $xy$.

Another nonlinear example

$\frac{dx}{dt} = e^x + y$

This is not linear because $e^x$ is not linear in $x$.

Being able to recognize linearity is a key skill because many methods in this unit only work for linear systems.

Solving linear systems: the main idea

The goal of solving a system is to find functions $x(t)$ and $y(t)$ that satisfy every equation at once. For linear systems, there are several major strategies.

One approach is to convert the system into a single higher-order equation. For example, from

$\frac{dx}{dt} = 3x - 2y$

$\frac{dy}{dt} = 4x + y$

you could differentiate one equation and use the other to eliminate one variable. This can produce a second-order differential equation in just one unknown function. That method is useful for small systems.

Another approach is to use matrix methods. Writing the system as

$\mathbf{x}' = A\mathbf{x}$

suggests looking for solutions of the form

$\mathbf{x}(t) = \mathbf{v}e^{rt}$

where $r$ is a constant and $\mathbf{v}$ is a vector. Substituting gives

$ r\mathbf{v}e^{rt} = A\mathbf{v}e^{rt}$

and since $e^{rt} \neq 0$, this becomes

$A\mathbf{v} = r\mathbf{v}$

This is an eigenvalue equation. The values $r$ are eigenvalues of $A$, and the vectors $\mathbf{v}$ are eigenvectors.

You do not need to master eigenvalues yet to understand the basic idea of linear systems, but this shows why matrix form is so important. It is the bridge between differential equations and linear algebra.

A worked example

Let’s study the system

$\frac{dx}{dt} = x + y$

$\frac{dy}{dt} = 2x - y$

First, check that it is linear. Both $x$ and $y$ appear only to the first power, and there are no products like $xy$.

In matrix form, this is

$\mathbf{x}$' = $\begin{pmatrix} 1$ & 1 \ 2 & -$1 \end{pmatrix}$$\mathbf{x}$

To understand the behavior of this system, we can look for solutions of the form $\mathbf{x}(t)=\mathbf{v}e^{rt}$. Then $r$ and $\mathbf{v}$ must satisfy

$\begin{pmatrix} 1$ & 1 \ 2 & -$1 \end{pmatrix}$$\mathbf{v}$ = r$\mathbf{v}$

This means the long-term behavior of the system is controlled by the eigenvalues of the matrix. If one eigenvalue is positive, solutions may grow; if negative, they may decay. If eigenvalues are complex, solutions may oscillate as they grow or shrink.

Even without carrying out every eigenvalue calculation here, you can already see the main message: linear systems let us study multiple changing quantities together using structure, not guesswork.

Connection to the larger topic of systems of differential equations

Linear systems are a special and very important part of systems of differential equations. They provide the foundation for later ideas in the topic, including:

matrix form,
eigenvalues and eigenvectors,
classification of solution behavior,
stability near equilibrium points.

The reason linear systems are studied so carefully is that they are both useful on their own and a starting point for more advanced ideas. Many nonlinear systems are approximated by linear systems near equilibrium points. This means that understanding linear systems helps you understand the behavior of more complicated systems too.

In a broader sense, linear systems teach you how several changing quantities can interact in a structured way. That skill appears across mathematics, physics, biology, economics, and engineering.

Conclusion

students, a linear system of differential equations is a system where the unknown functions and their derivatives appear in a linear way. It can be written neatly in matrix form as $\mathbf{x}' = A\mathbf{x}$ for homogeneous systems, or $\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t)$ for nonhomogeneous systems. Linear systems are important because they model interacting quantities, support powerful methods like superposition and matrix analysis, and connect directly to eigenvalues and eigenvectors. This lesson is the starting point for understanding how systems of differential equations work as a whole 🎯.

Study Notes

A system of differential equations contains more than one differential equation with more than one unknown function.
A linear system is one in which the unknown functions appear only to the first power and are not multiplied together.
A system is homogeneous if it has no extra forcing term, so it can be written as $\mathbf{x}' = A\mathbf{x}$.
A system is nonhomogeneous if it has a forcing term, so it can be written as $\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t)$.
Matrix form organizes a linear system into a compact and useful structure.
The form $\mathbf{x}(t)=\mathbf{v}e^{rt}$ leads to the eigenvalue equation $A\mathbf{v}=r\mathbf{v}$.
Eigenvalues and eigenvectors help describe how solutions grow, decay, or oscillate.
Linear systems are important because they model many real situations and serve as the foundation for more advanced study in systems of differential equations.