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If you find yourself in a hole, stop digging, Will Rogers
I’d far rather be happy than right any day, Douglas Adams, The Hitchhiker’s Guide to the Galaxy
Definition. A differential equation is an equation that involves one or more dependent variables, their derivatives with respect to one or more independent variables, and the independent variables themselves, e.g., $\frac{dy}{dx} = 3x +5y, y’ + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$
It involves (e.g., $\frac{dy}{dx} = 3x +5y$):
Definition. A first-order linear ordinary differential equation is an ordinary differential equation (ODE) involving an unknown function y(x), its first derivative y′, and functions of the independent variable x, which can be written in the general form:: a(x)y' + b(x)y = c(x) where:
These equations are termed “linear” because the unknown function y and its derivative y’ appear to the first power and are not multiplied together or composed in any nonlinear way.
If the function c(x)=0 for all x in the interval of interest, the equation simplifies to: a(x)y’ + b(x)y = 0. Such an equation is called a homogeneous linear differential equation.
The Existence and Uniqueness Theorem provides crucial insight into the behavior of solutions to first-order differential equations ODEs. It states that if:
Then, the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x0, y0), meaning that it satisfies the initial condition y(x0) = y0.
This theorem ensures that under these conditions, the solution exists and is unique near x = x0.
Non-linear autonomous systems describe how the state of a system evolves over time when the system’s rules do not explicitly depend on time. By focusing on two variables x and y, we have a non-linear autonomous system:
$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$
This system defines how x(t) and y(t) change with respect to time t. The functions f and g describe the system’s “velocity” in the x and y directions, respectively.
By transforming a non-linear autonomous system into a single first-order ordinary differential equation (ODE), we can often simplify the problem and gain deeper insights into the system’s behavior. This approach allows us to analyze complex systems using methods developed for single-variable differential equations.
$\begin{cases} \frac{dx}{dt} = f(x, y) ~i\\ \frac{dy}{dt} = g(x, y) ~ii\end{cases}$
To better understand the behavior of this system, we can construct the velocity field $\vec{F}$ which gives a geometric interpretation of the system. It provides a visual representation of how the system evolves over time. Each vector indicates the direction and speed at which a point (x, y) moves.
The velocity field is defined as: $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$.
This vector field describes how the values of x and y change over time at every point in the plane. Solutions to the system are pairs of functions x(t) and y(t), but geometrically, they are trajectories, paths or integral curves traced out by the evolving system in the xy-plane. These trajectories are tangent to the vector field $\vec{F}$ at every point, meaning the direction of motion at any point matches the vector field’s direction.
To simplify the analysis, we can eliminate the parameter t from the system. By dividing equation (ii) by equation (i), we obtain: $\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{g(x, y)}{f(x, y)}⇒\frac{dy}{dx} = \frac{g(x, y)}{f(x, y)}$.
We convert the original non-linear autonomous system into a single first-order differentiable equation (ODE) that might be solvable.
Originally, the system provides a velocity field $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ indicating the direction and speed at which a point (x, y) moves over time. By converting it into the ODE $\frac{dy}{dx} = \frac{g(x, y)}{f(x, y)}$, we shift our focus to a slope field (involving only x and y). Here, we’re interested in the slope of the trajectories at each point in the plane, rather than their velocities over time (Refer to Figure i for a visual representation and aid in understanding it).
The solutions to this ODE, whether explicit y = y(x) or implicit h(x, y)=0, represent integral curves. These curves are paths in the xy-plane that have a slope at each point matching $\frac{dy}{dx}$. By solving this ODE, we can find the trajectories of the system without directly involving the time parameter t.
Step 1: Eliminating the Parameter t
To simplify the system and analyze the trajectories in the xy-plane, we can eliminate the parameter t by finding $\frac{dy}{dx}$.
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}=\frac{y’}{x’} =[\text{Substituting the expressions from equations (1) and (2):}] \frac{x + y^2}{x^2−y}$
Thus, we obtain the first-order ODE: $\frac{dy}{dx} = \frac{x + y^2}{x^2−y}$
Our next goal is to solve this ODE to find the trajectories y(x). However, this ODE is non-linear and not readily solvable using standard methods, we can perform a qualitative analysis to understand the system’s behavior.
Step 2: Qualitative Analysis.
Finding Critical Points Critical points occur where x’ = 0 and y’ = 0
$\begin{cases} x^2−y = 0 \\ x + y^2 = 0 \end{cases}$
$x^2−y = 0 ⇒ y = x^2$. Substitute this result into the second equation: $x + y^2 = x + x^4 = 0↭[\text{Factor out x}] x(1 + x^3) = 0 ↭[\text{Solve for x}] x = 0, x = -1$. For x = 0, y = 02 = 0. For x = -1, y = (-1)2 = 1. Therefore, critical points: (0, 0), (-1, 1).
Linearization Around Critical Points
We can linearize the system near each critical point to analyze stability.
Compute the Jacobian matrix $J = (\begin{smallmatrix}\frac{∂f}{∂x} & \frac{∂f}{∂y}\\ \frac{∂g}{∂x} & \frac{∂g}{∂y}\end{smallmatrix}) = (\begin{smallmatrix}2x & -1\\ 1 & 2y\end{smallmatrix})$
At (0, 0): $J = (\begin{smallmatrix}0 & -1\\ 1 & 0\end{smallmatrix})$
Eigenvalues λ: det(J -λI) = $det(\begin{smallmatrix}-λ & -1\\ 1 & -λ\end{smallmatrix}) = λ^2 + 1 ⇒ λ = ±i$. The eigenvalues are purely imaginary, indicating that the critical point (0,0) is a center. Nearby trajectories are closed orbits around this point. However, one must be careful, as non-linear terms can change this conclusion. One must do further analysis (or rely on known theorems or simulations) to confirm the existence or absence of a limit cycle.
At (-1, 1): $J = (\begin{smallmatrix}-2 & -1\\ 1 & 2\end{smallmatrix})$
Eigenvalues λ: det(J -λI) = $det(\begin{smallmatrix}-2-λ & -1\\ 1 & 2-λ\end{smallmatrix}) = (−2−λ)(2−λ)−(−1)(1)=(λ+2)(λ−2)−(−1)= λ^2-3= 0 ⇒λ = ±\sqrt{3}$.
The eigenvalues are real and of opposite signs, hence the critical point (−1, 1) is a saddle point. Trajectories near this point leave or move away along the unstable manifold one direction and approach or move towards the stable manifold along another. A saddle point is typically unstable.
Saddle points do not by themselves create closed orbits. Instead, they often serve as boundaries between different types of trajectory behaviors. This qualitative analysis allows us to understand the dynamics of the system without explicitly solving the ODE. This approach is widely used in mechanics, biology (e.g., predator-prey models), and other fields of science and engineering to analyze complex systems’ behavior.
We start with the system of differential equations that describes a simple harmonic oscillator: $\begin{cases} x’ = y \\ y’ = -x \end{cases}$. Here, x and y are functions of t, and the primes denote derivatives with respect to t.
The general solution is $(\begin{smallmatrix}x\\ y\end{smallmatrix}) = c_1(\begin{smallmatrix}cos(t)\\ -sin(t)\end{smallmatrix})+c_2(\begin{smallmatrix}sin(t)\\ cos(t)\end{smallmatrix})$ where c1 and c2 are constants determined by initial conditions.
Each component of the solution is a combination of sines and cosines. This is a hallmark of harmonic oscillators, whose solutions are naturally periodic functions. From the form of the solution, we can see that as t varies, (x(t), y(t)) traces out a circle.
These equations can be thought of as describing a velocity field (x’ , y′) = (y,−x). At any point (x, y), the vector (y, −x) is perpendicular to the radius vector (x, y) drawn from the origin. In other words, the motion is always “tangential” to the circle centered at the origin.
This system describes a simple harmonic oscillator, and its solutions are a family of concentric circles centered at the origin (circular trajectories) , representing periodic motion. Each trajectory is a closed curve, and the motion goes around clockwise indefinitely, which is typical of simple harmonic oscillators. The constant coefficients c1 and c2 determine the specific circle (its radius and starting point).
The motion is periodic, and each initial condition determines the circle’s radius and the starting position on that circle.
Eliminating t from the system, we get:
$\frac{dy}{dx} = \frac{y’}{x’} = \frac{-x}{y}$. TThis is a separable differential equation, which can be solved as follows:
This equation represents a family of circles centered at the origin. The constant 2C corresponds to the square of the radius of each circle. This confirms the geometric interpretation of the original system’s solutions as circular trajectories.
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