Calculus

Intro to Differential Equations

A differential equation is an equation that involves a function and its derivatives. Instead of saying “y equals something,” it says “the rate of change of y equals something.” Your job is to find the function that satisfies that rule.

This is where calculus meets the real world. Population growth, radioactive decay, cooling coffee, spreading diseases — they are all described by differential equations.

1. The Simplest Example

Start with the simplest differential equation imaginable:

\frac00 = k

This says: the slope is constant everywhere. What function has a constant slope? A straight line: y = kx + C, where C is the starting value.

k (constant slope)1

-33

C (initial value)0

-55

\frac00 = 1 \implies y = 1 \cdot x + 0

Try This

Try this: Change C and watch the line shift up or down — different starting points, but the same slope everywhere. Change k and the line tilts. Each combination of k and C gives a different solution, but they all satisfy the same differential equation.

2. Exponential Growth: dy/dx = ky

Here is the equation that governs population growth, compound interest, and radioactive decay:

\frac00 = k \cdot y

This says: the rate of change of y is proportional to y itself. The bigger y is, the faster it grows (or shrinks). The solution is an exponential:

y = y_0 \cdot e^0

k (growth/decay rate)0.5

-22

y0 (initial value)1

0.55

\frac00 = 0.5 \cdot y, \quad y(0) = 1 \implies y = 1 \cdot e^{{0.5 x}}

Connection

When k > 0, you get exponential growth — the function and its derivative both increase without bound. When k < 0, you get exponential decay — the function approaches zero but never quite reaches it. The red curve (the derivative) is always exactly k times the blue curve. That is what the DE says.

3. Slope Fields: Visualizing the DE

A slope field is a grid of tiny line segments showing the slope dy/dx at each point. It is like a map of wind directions — the slope field shows you which way the solution curves will flow.

For dy/dx = -x/y (which describes circles), every point (x, y) has a slope of -x/y.

Let’s visualize the slope field for a simpler equation: dy/dx = x.

C (initial condition: y(0) = C)0

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\frac00 = x \implies y = \frac{x^2}0 + C

Try This

Try this: The red line shows the slope function (dy/dx = x). Notice that all the parabolic solution curves have the same shape — they are just shifted up and down by the constant C. At any x-value, every solution curve has the same slope. That is because the slope depends only on x, not on y.

4. When the Slope Depends on y

Things get more interesting when the slope depends on y. Consider:

\frac00 = -y \quad \implies \quad y = y_0 \cdot e^{-x}

This is exponential decay. Every solution curve decays toward zero, but the rate of decay depends on the current value of y.

y0 (initial condition)3

0.55

Connection

All the curves have the same shape but different scales. A larger initial value means a steeper initial descent (because dy/dx = -y, and y is larger). But all curves converge toward zero. In the real world, this models anything that decays proportionally to its current amount: radioactive isotopes, cooling temperatures, drug concentrations.

5. The Role of Initial Conditions

A differential equation typically has infinitely many solutions — a whole family of curves. The initial condition y(0) = y0 picks out one specific curve from the family.

\frac00 = \cos(x) \implies y = \sin(x) + C

C (choose a solution)0

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Try This

Try this: Slide C to see different members of the solution family. All curves have the same shape (they are all sine waves) but are shifted vertically. The red curve (the slope function cos(x)) is the same for all of them — it only depends on x. The initial condition C pins down which specific sine wave you are on.

Challenge

Challenge: A population P grows according to dP/dt = 0.03P (3% growth rate). If the initial population is P(0) = 1000, write the solution. How long does it take for the population to double? Set the solution equal to 2000 and solve for t.

Hint: ln(2) / 0.03 gives you the doubling time.

The Big Idea

A differential equation describes how a quantity changes. Solving it means finding the function whose rate of change matches the rule. The initial condition picks one solution from infinitely many.

Differential equations are the language of change in the natural world. Newton’s laws of motion, Maxwell’s equations of electromagnetism, the heat equation, population models — they are all differential equations. Learning to read and solve them is learning to decode how nature works.

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