Antiderivatives and differential equations

So far you have started with a function and found its derivative. Engineering problems often run the other way. An accelerometer measures acceleration, and you want velocity and position. A flow meter measures the rate at which water enters a tank, and you want the volume. A physical law tells you how fast a temperature or a voltage changes, and you want the temperature or voltage itself. Undoing differentiation is called antidifferentiation, and equations that relate a function to its derivatives are called differential equations. This lesson introduces both: how to find antiderivatives, how initial conditions pin down a particular answer, how to solve the simplest and most important differential equations of growth, decay, cooling and circuit response, how to picture solutions with slope fields, and how to compute them numerically with Euler's method when no formula is available. The definite integral of Lesson 9 builds directly on these ideas.

Antiderivatives

A function \(F\) is an antiderivative of \(f\) on an interval if \(F'(x) = f(x)\) for every \(x\) in the interval. For example, \(F(x) = x^3\) is an antiderivative of \(f(x) = 3x^2\). So are \(x^3 + 5\) and \(x^3 - \pi\), because constants differentiate to zero. Lesson 6 showed that two functions with the same derivative on an interval differ only by a constant, so every antiderivative of \(f\) has the form \(F(x) + C\). This family is written with the indefinite integral sign:

where \(C\) is an arbitrary constant, the constant of integration. Each differentiation rule, read backward, gives an antiderivative rule:

Antiderivatives of sums and constant multiples follow term by term. Always check an antiderivative by differentiating it: you should get back the original function.

Not every function has an antiderivative that can be written with elementary functions. The antiderivative of \(e^{-x^2}\), which is central to probability, exists but cannot be expressed as a finite combination of powers, exponentials, logarithms and trigonometric functions. Such antiderivatives are computed numerically or defined as new special functions.

Initial value problems

An antiderivative contains an unknown constant. One known value of the function, an initial condition, fixes it. A problem of the form "find \(y\) such that \(y' = f(x)\) and \(y(x_0) = y_0\)" is an initial value problem.

Motion from acceleration

The most familiar initial value problem is motion with known acceleration. Since \(v'(t) = a(t)\) and \(s'(t) = v(t)\), velocity is an antiderivative of acceleration, and position is an antiderivative of velocity. Two initial conditions, the starting velocity \(v_0\) and position \(s_0\), fix the two constants. For constant acceleration \(a\):

These are the kinematic equations of introductory physics (PHYS-130), derived here in two lines of calculus.

Differential equations

A differential equation relates an unknown function to one or more of its derivatives. The equations above, \(y' = f(x)\), are the simplest kind, in which the derivative is given directly. More interesting equations involve the unknown function itself: Newton's second law for a spring, \(m x'' = -kx\); the cooling law \(T' = -k(T - T_a)\); the circuit law \(RC \, V' + V = V_s\). A solution is a function that satisfies the equation identically. The order of the equation is the highest derivative it contains.

Checking a proposed solution needs only differentiation. For example, \(x(t) = \cos(\omega t)\) with \(\omega = \sqrt{k/m}\) satisfies \(m x'' = -kx\), because \(x'' = -\omega^2 \cos \omega t = -(k/m) x\). The general theory of solving differential equations is the subject of a later course (MATH-201); here we solve the most important first-order cases.

Exponential growth and decay

The simplest differential equation involving the function itself says that a quantity changes at a rate proportional to its size:

To solve it, divide by \(y\) (assuming \(y \ne 0\)) to get \(\dfrac{y'}{y} = k\). The left side is the derivative of \(\ln|y|\), so \(\ln|y| = kt + C\), and exponentiating gives \(y = y_0 e^{kt}\), where \(y_0 = y(0)\). This is the exponential model of Lesson 2, now derived from the rate law that underlies it: radioactive decay, continuously compounded interest, unconstrained population growth, the discharge of a capacitor through a resistor and first-order chemical reactions all obey it.

Approach to an equilibrium

Many systems relax toward an equilibrium value at a rate proportional to their distance from it. Newton's law of cooling states that an object's temperature changes at a rate proportional to the difference between it and the surroundings:

Substituting \(u = T - T_a\) gives \(u' = -ku\), so \(u = u_0 e^{-kt}\) and:

The same equation describes a capacitor charging through a resistor from a source \(V_s\): \(RC\,\dfrac{dV}{dt} = V_s - V\), with solution \(V(t) = V_s(1 - e^{-t/RC})\) when the capacitor starts uncharged. It also describes a thermometer settling to a new reading, a tank level approaching equilibrium through an outlet, and the speed of a lightly loaded motor approaching its final value. In each case, the response reaches about 63 percent of its final change after one time constant and about 99 percent after five.

Slope fields

A first-order differential equation \(y' = f(t, y)\) gives the slope of the solution at every point of the plane. Drawing short line segments with these slopes at a grid of points produces a slope field (or direction field), which shows the shape of all solutions at once. Each solution curve follows the segments, like a leaf floating on a stream. Slope fields reveal qualitative behavior without solving the equation: equilibrium solutions (constant solutions, where \(y' = 0\)), whether solutions approach or move away from them, and how fast.

In the cooling equation, \(T = T_a\) is an equilibrium, and every solution approaches it: a stable equilibrium. In the growth equation \(y' = ky\) with \(k > 0\), the equilibrium \(y = 0\) is unstable: any solution that starts slightly away from it moves farther away. A more realistic population model, the logistic equation \(y' = ry(1 - y/K)\), has an unstable equilibrium at 0 and a stable one at the carrying capacity \(K\), and its slope field shows S-shaped solution curves rising from small populations to \(K\).

Euler's method

Most differential equations that arise in practice cannot be solved with a formula. They are solved numerically, and the simplest numerical method, due to Leonhard Euler, follows the slope field in small steps. Starting from the initial point \((t_0, y_0)\), take a step of size \(h\) along the tangent line, whose slope is \(f(t_0, y_0)\), and repeat:

Each step is a linear approximation (Lesson 3), and each introduces a small error. Over a fixed interval, the total error of Euler's method is roughly proportional to \(h\): halving the step size roughly halves the error. More accurate methods, such as the Runge-Kutta methods used in engineering software, sample the slope at several points within each step, but they rest on the same idea.

Engineers rarely write their own differential equation solvers, but understanding Euler's method explains the behavior of the solvers in simulation software: why step size matters, why results can drift, and why some equations, called stiff equations, require special methods to avoid instability. Lesson 11 returns to numerical methods.