From a context to an equation

A differential equation links an unknown function to one or more of its derivatives. Instead of giving the quantity directly, it describes how that quantity changes.

When building one from a context, identify:

• the dependent quantity;
• the independent quantity, often time;
• the rate being described;
• words such as “proportional to”, “inversely proportional to” or “at a constant rate”.

Suppose, for example, that the rate at which a medicine leaves the bloodstream is proportional to the amount present:

The negative sign is essential because the amount is decreasing. Also, “proportional to” introduces a constant; it doesn’t mean that the two quantities are equal without one.

Check the units. Here $dQ/dt$ has units $\text{kg s}^{-1}$, as does $kQ$. This quick check often reveals an incorrect model.

Separation of variables

A separable differential equation is a first-order differential equation that can be rearranged so every expression involving the dependent variable sits with its differential, while every expression involving the independent variable is placed on the other side.

Consider the general equation

Separate the variables before integrating:

Now integrate both sides. Include a constant of integration. Only one is needed, since two arbitrary constants can be combined.

For the important proportional-growth model

separating the variables gives

Therefore,

After exponentiating, absorb the sign into a new arbitrary constant:

The result is an exponential family of solutions.

A general solution is a family of functions with enough arbitrary constants to represent all solutions admitted by the differential equation on the interval being considered. A first-order equation normally gives one arbitrary constant. If an initial condition such as $y(x_0)=y_0$ is provided, substitute it to find the constant and obtain a particular solution. This is one member of the family that satisfies the given condition; $x_0$ is the specified initial value of $x$ and $y_0$ is the corresponding specified value of $y$, with their respective context-dependent units.

Take care when dividing by an expression involving $y$. If the expression can equal zero, test the resulting constant function separately. Otherwise, the division may discard a valid equilibrium solution.

Reading the local slopes

A slope field is a diagram made up of short line segments. At each plotted point, the segment’s gradient matches the derivative given by a first-order differential equation. For

The field doesn’t show a single solution. Instead, it gives the local direction that every solution must follow. To sketch a solution curve, move smoothly through the field and keep the curve tangent to the segments. Different initial conditions can be traced without solving the equation algebraically.

When reading a field, look for:

• horizontal segments, where $f(x,y)=0$;
• regions with positive or negative gradient;
• places where the magnitude of the gradient becomes large;
• horizontal solution curves that act as equilibrium solutions;
• repeated slope patterns along horizontal, vertical or diagonal bands.

In a well-behaved first-order model, solution curves through different initial points don’t normally cross. If they crossed, the equation would assign two different solution directions to the same point.

Representation and choice

A slope field is one chosen representation of a differential equation, not the equation itself. What the viewer notices depends on the window, scale, density of segments and chosen sample points. A coarse field might hide an equilibrium or make rapid change look mild. This shows how values and choices influence mathematical representations of the world: the underlying rule may stay fixed, but the diagram still reflects human decisions about what to display.

One first-order equation

A numerical solution gives a sequence of approximate values from a differential equation and initial data when an exact formula is unavailable or not needed. In Euler’s method, the solution follows a tangent line for one short step. The slope is then recalculated.

For $dy/dx=f(x,y)$, the Euler update is

The rule is

For each row, use the current row to calculate the slope—not the new row. That detail defines Euler’s method.

Using a smaller $|h|$ usually improves the approximation because the tangent line is followed across a shorter interval. It also requires more steps and still doesn’t make the result exact. A positive $h$ moves forward in $x$; a negative $h$ gives a backward approximation. Error builds up through the local straight-line approximation and numerical rounding.

A spreadsheet handles this recurrence well. Put $x_n$, $y_n$, $f(x_n,y_n)$, $x_{n+1}$ and $y_{n+1}$ in columns, enter the initial row, then fill the formulas downward. In an examination, permissible technology may generate the values directly, but the recurrence shows what the technology is doing.

Coupled first-order systems

A coupled differential system contains differential equations in which each state variable’s rate can depend on several state variables at the same time. For example,

Euler’s method updates both variables using the same current state:

Spreadsheet-style Euler updates for a coupled system.

Coupled models can describe two interacting populations, such as a consumer population and its food source. Lotka–Volterra models form a standard family of these interaction models. The same numerical method also applies to an SIR infection model, where susceptible, infectious and recovered groups are updated together.

Once the initial conditions, model, step size and rounding rules have been fixed, the Euler values are mathematically determined. Certainty about the calculated values, though, doesn’t guarantee certainty about the real system. Parameters may be estimated, assumptions may be unrealistic, and discretization introduces approximation. More advanced schemes, including Runge–Kutta methods, usually achieve greater accuracy per step by sampling rates more carefully.

Linear systems in the phase plane

Consider the linear autonomous system

The matrix form is

The notation ${}^{\mathsf T}$ denotes transpose.

A phase plane is a coordinate plane with the state variables on its axes instead of time. One solution traces out a curve in this plane as time changes; this curve is called a trajectory. A phase portrait shows a collection of representative trajectories, with arrows indicating the direction of increasing time.

Because time isn’t an axis, these direction arrows matter. At any point $(x,y)$, the system gives the velocity vector $(dx/dt,dy/dt)$, which determines the local direction of travel.

An equilibrium point is a state where all rates of change are zero. Find it by solving

The eigenvalues in the systems considered here are distinct and non-zero, making the origin the relevant isolated equilibrium. In a population model, a different translated equilibrium could represent stable population levels.

Eigenvalues and long-term behaviour

An eigenvector is a non-zero vector whose direction stays unchanged when it is multiplied by a matrix. The scalar factor by which that vector is multiplied is its eigenvalue. The relationship is

The type and sign of the eigenvalues determine how paths behave in the future:

Positive real parts lead to motion away from the origin; negative real parts lead to motion towards it. Complex eigenvalues give spiralling paths unless the real part is zero. If the real eigenvalues have opposite signs, the result is a saddle point: an equilibrium approached along one characteristic direction and left along another.

Phase-portrait types matched to eigenvalue patterns.

To sketch a phase portrait, mark the equilibria first. Use the eigenvalues to determine the shape and stability, then add arrows. For real eigenvalues, draw the eigendirections accurately, since they guide the trajectories. Context also matters. In population models, for example, negative coordinates may be mathematically valid but have no meaning.

Exact solutions for distinct real eigenvalues

Exact calculation is required only when the eigenvalues are real and distinct. If they are $\lambda_1$ and $\lambda_2$, with corresponding eigenvectors $\mathbf{v}_1$ and $\mathbf{v}_2$, then

The initial conditions determine $C_1$ and $C_2$.

For a non-linear system, a Jacobian matrix provides a local linear approximation near an equilibrium and can be used to investigate its stability. This gives useful wider context, although the systems considered here are already linear.

Converting to a first-order system

A second-order differential equation has a second derivative as its highest derivative. Its general time-dependent form is

Now introduce the velocity variable

The original equation can then be written as the coupled first-order system

Once the equation has been converted, Euler’s method for coupled equations applies directly.

For time step $h$, use

The values $X_{n+1}$ and $V_{n+1}$ are the next approximate position and velocity. Both updates must use the old row. A spreadsheet can generate the successive values efficiently.

Linear homogeneous equations

One common form is

The same structure is often written with other constant letters, such as $a$ and $b$.

Setting $v=dx/dt$ gives

The equation can now be studied using the eigenvalue and phase-portrait methods for coupled systems. If the eigenvalues are distinct and real, the exact solution comes from the two corresponding eigenvectors and exponential terms. The initial position and initial velocity determine the two arbitrary constants.

In the phase plane, one axis represents position and the other represents velocity. Inward trajectories show damping. Closed loops show sustained periodic motion in an idealized undamped model, while outward trajectories show growing oscillations or instability.

Second-order equations arise naturally in physical phenomena because acceleration is the second derivative of position. They can model motion with restoring forces, resistance or both. In an examination, the physical equation will be supplied. The task is to convert, approximate, solve or interpret it, rather than derive the governing law from physics.

Euler’s historical influence shows how individuals can shape mathematical knowledge. His named method turns a geometric tangent idea into a repeatable numerical procedure. However, the result doesn’t depend on authority: the recurrence can be justified, tested and improved independently of its name.