Models, Direction Fields, and Solution Curves

Most laws of nature are statements about rates. Newton's second law relates the acceleration of a body to the forces on it; the rate a population grows is tied to its current size; heat leaves an object at a rate set by how much hotter it is than its surroundings. Acceleration and growth rate and cooling rate are all derivatives, so writing a law of this kind in symbols produces an equation that contains derivatives of an unknown function. That is a differential equation.¹

The first-order case has the form

\frac{d y}{d t} = f (t, y),

where $f$ is a given function of the independent variable $t$ and the dependent variable $y$ . Such an equation assigns a slope to every point of the plane, and that assignment alone determines how solutions behave, often before one can be written down.

Building a model

A differential equation that describes a physical process is a mathematical model of it.² Deriving one is a fixed sequence: name the variables, state the governing principle, and translate the principle into symbols.

A falling body

Suppose an object of mass $m$ falls near the earth's surface. Let $t$ be time and $v (t)$ its velocity, positive downward. Newton's second law says mass times acceleration equals net force, and acceleration is $d v / d t$ :

m \frac{d v}{d t} = F .

Two forces act. Gravity pulls down with magnitude $m g$ , where $g \approx 9.8 m/ s^{2}$ . Air resistance opposes the motion; for a slowly falling object it is well approximated as proportional to speed, with magnitude $γ v$ for a drag coefficient $γ > 0$ . Gravity acts in the positive direction and drag in the negative, so the net force is $m g - γ v$ and the model is

m \frac{d v}{d t} = m g - γ v .

The constants split into two kinds. The mass $m$ and drag coefficient $γ$ are parameters: they change from one object to another. The gravitational acceleration $g$ is a physical constant, the same for every object. Taking $m = 10 kg$ and $γ = 2 kg/s$ gives a concrete instance,

\frac{d v}{d t} = 9.8 - \frac{v}{5} .

A cooling object

Newton's law of cooling states that the temperature $u (t)$ of an object changes at a rate proportional to the difference between its temperature and the ambient temperature $T$ of the surroundings:

\frac{d u}{d t} = - k (u - T), k > 0.

The sign is arranged so that an object hotter than its surroundings ( $u > T$ ) cools ( $d u / d t < 0$ ) and a colder one warms. The same equation governs both cases.

A population under predation

Let $p (t)$ be the size of a population that, left alone, grows at a rate proportional to its current size, the rate constant $r$ absorbing birth and death rates. If a predator removes members at a fixed rate $k$ , the model is

\frac{d p}{d t} = r p - k .

All three models share a form. Writing the dependent variable as $y$ ,

\frac{d y}{d t} = a y - b,

with constants $a$ and $b$ : the falling body is the case $a = - γ / m$ , $b = - g$ ; cooling is $a = - k$ , $b = - k T$ ; the population is $a = r$ , $b = k$ . One analysis covers all three.

Reading the slope field

The rate function turns the equation into a geometric object. At each point $(t, y)$ of the plane, $f (t, y)$ is the slope a solution curve must have there. Drawing a short segment of that slope at each point of a grid produces a direction field (or slope field). A solution is then any curve that stays tangent to the segments as it threads through them.³

The construction never requires solving the equation. It evaluates $f$ on a grid and draws a segment at each point — work a computer does in an instant, even for equations no elementary method can solve.

Algorithm:

\textsc{DirectionField}(f, [t_0, t_1] \times [y_0, y_1], h)

— sample the slope field

1
for each grid point $(t, y)$ with $t \in [t_0, t_1]$ , $y \in [y_0, y_1]$ stepped by $h$ do
2
$m \gets f(t, y)$
the slope the equation demands here
3
draw a short segment centered at $(t, y)$ with slope $m$
4
end

For the falling body $d v / d t = 9.8 - v /5$ , the slope is $1.8$ along the line $v = 40$ , drops to $- 0.2$ along $v = 50$ , and reaches $- 2.2$ along $v = 60$ . Below some critical velocity every segment tilts upward and the object speeds up; above it every segment tilts down and the object slows. The generic form $d y / d t = a y - b$ shows the same split around a single horizontal line.

Slope field of

d y / d t = 2 - y

: segments flatten toward the line

y = 2

, and every solution curve bends to follow them onto it.

Four conclusions come straight off that picture, none of them requiring a formula:

A special constant solution exists. Where the slope is zero the segments are horizontal, and the horizontal line through them is itself a solution.
The other solutions bend toward it. Below the line slopes are positive and curves rise; above it slopes are negative and curves fall.
Behavior is set by starting position, not by the exact start. Two curves starting close together end up close together, both approaching the line.
The long-run value is the same for all of them. As $t$ grows, every solution converges to $y = 2$ .

Equilibrium solutions

The horizontal line in the figure is an equilibrium.

For the falling body, setting $9.8 - v /5 = 0$ gives $v = 49 m/s$ . The constant function $v (t) = 49$ satisfies the equation: substitute it and both sides are zero. Physically it is the velocity at which gravity and drag balance, the terminal velocity, and every other solution approaches it. In general $d y / d t = a y - b$ has the single equilibrium $y = b / a$ .

Not every equilibrium attracts. The two behaviors are distinguished by the sign of the rate function on either side of the constant line.

An attracting equilibrium (left,

a < 0

) pulls neighboring solutions in; a repelling one (right,

a > 0

) pushes them away. The line is a solution in both panels.

The falling body and cooling object have $a < 0$ and attracting equilibria: left alone long enough, they settle. The predation model has $a = r > 0$ and a repelling one at $p = k / r$ : a population starting above it grows without bound, one starting below it dwindles to extinction. The equilibrium is the boundary between those two outcomes, observed in practice only if the population begins exactly on it. This attracting-versus-repelling distinction generalizes into stability theory, which classifies equilibria systematically.

	Attracting (stable)	Repelling (unstable)
Slope of $f$ at equilibrium	$f^{'} (y^{*}) < 0$	$f^{'} (y^{*}) > 0$
Nearby solutions	converge to $y^{*}$	diverge from $y^{*}$
Falling body / cooling	terminal velocity, ambient temp.	—
Population under predation	—	critical population $k / r$
Observed after long time	the equilibrium itself	any value but the equilibrium

The one-dimensional summary is the phase line: the $y$ -axis alone, marked with the equilibria and with arrows giving the sign of the rate function between them. For the falling body it carries a single attracting point.

Phase line of the autonomous falling body

d v / d t = 9.8 - v /5

. The single equilibrium

v = 49

attracts from both sides — the rate

f

is positive below it and negative above it — so every solution slides onto terminal velocity.

Solving the linear model

The equation $d y / d t = a y - b$ can be solved outright, confirming every conclusion drawn from the slope field.⁴

The same steps applied to $d y / d t = a y - b$ produce

y (t) = \frac{b}{a} + c e^{a t},

the equilibrium $b / a$ plus a transient $c e^{a t}$ that decays when $a < 0$ and grows when $a > 0$ , exactly the attracting and repelling cases of the figure.

Every solution of a first-order equation carries one arbitrary constant because solving involves one integration, and each integration brings a constant with it.⁵ The family fills the strip of the plane where the model applies, one curve through each point.

Integral curves

p = 900 + c e^{t /2}

d p / d t = 0.5 p - 450

. The equilibrium

c = 0

divides curves that rise from those that fall; the marked curve is the one with

p (0) = 850

Initial conditions and the initial value problem

A single solution is picked out by naming a point the curve must pass through. That extra requirement is an initial condition, and the equation together with it is an initial value problem.

	General solution	Particular solution
Arbitrary constant	free ( $c$ ranges over $R$ )	fixed by $y (t_{0}) = y_{0}$
Graph	family of integral curves	one curve through $(t_{0}, y_{0})$
Example	$p = 900 + c e^{t /2}$	$p = 900 - 50 e^{t /2}$

Whether such a problem has a solution, and only one, is the existence and uniqueness question. For rate functions as well-behaved as these the answer is yes, and the picture already assumed it: one integral curve through each point. The precise hypotheses, and the equations where uniqueness fails, wait for a later lesson.

The cooling family and a worked case

The cooling model $d u / d t = - k (u - T)$ has the same structure, with equilibrium $u = T$ and $a = - k < 0$ , so every solution decays to ambient temperature. Its general solution is

u (t) = T + (u_{0} - T) e^{- k t},

an exponential approach to $T$ from whatever the initial temperature $u_{0}$ was.

Cooling curves

u = 20 + (u_{0} - 20) e^{- t}

for several starting temperatures. Objects hotter and colder than the ambient

20

both relax onto it, the difference shrinking exponentially.

Limitations of the models

A model is only as good as its assumptions, and each of these has a limited range.² The linear drag law holds for small, slow objects; a fast one needs a drag force proportional to $v^{2}$ , a nonlinear term that changes the analysis. The predation model predicts unbounded growth above the equilibrium and negative populations below it, both physically impossible past a short horizon; the logistic equation bends growth down as crowding sets in. These first models are worth studying for the method rather than their numbers: derive the equation from a principle, read the slope field for equilibria and long-run behavior, then solve when the equation is simple enough and fix the constant with an initial condition.

Simmons, Differential Equations with Applications and Historical Notes, Ch. 1 §1 — Introduction: differential equations as the natural language of laws relating quantities to their rates of change, with the freely falling body as the opening example. ↩
Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems, §1.1 — Some Basic Mathematical Models: the falling-object and field-mouse derivations, the modeling steps, parameters versus physical constants, and the limitations of each model. ↩ ↩²
Boyce & DiPrima, §1.1 — Direction Fields: constructing the slope field by evaluating the rate function on a grid, and drawing qualitative conclusions (equilibria, convergence) without solving. ↩
Boyce & DiPrima, §1.2 — Solutions of Some Differential Equations: solving $d y / d t = a y - b$ by integration, the general solution $y = b / a + c e^{a t}$ , integral curves, the initial value problem, and the $300 m$ falling-object computation. ↩ ↩² ↩³ ↩⁴ ↩⁵
Simmons, Ch. 1 §2 — General Remarks on Solutions: verifying solutions by substitution, the appearance of arbitrary constants equal in number to the order, and the general versus particular solution. ↩