Separable and linear first-order differential equations

Calculus I introduced differential equations through the simplest cases: \(y' = f(x)\), solved by antidifferentiation, and \(y' = ky\), whose solutions are exponentials, together with Newton's law of cooling and Euler's method. With the integration techniques of Lessons 1 and 2, a much wider class of equations can now be solved exactly. This lesson covers the two most important families of first-order equations. Separable equations, in which the variables can be moved to opposite sides, describe tank draining, population growth with limits, chemical reactions and falling objects with air resistance. Linear equations, solved with an integrating factor, describe mixing tanks, electrical circuits with time-varying sources and the heating of objects whose surroundings change. Together they cover a large share of the first-order models used in engineering, and they prepare you for the systematic treatment in MATH-201.

Separable equations

A first-order differential equation is separable if it can be written as:

that is, the right side is a product of a function of \(x\) alone and a function of \(y\) alone. To solve it, separate the variables by moving everything involving \(y\) to one side and everything involving \(x\) to the other, then integrate both sides:

The result is an equation relating \(x\) and \(y\), with one constant of integration, which an initial condition then fixes. Solve for \(y\) explicitly where possible. The method is justified by the chain rule: if \(H\) is an antiderivative of \(1/h\), then \(\frac{d}{dx}H(y) = \frac{1}{h(y)}\frac{dy}{dx} = g(x)\).

Two cautions apply. Dividing by \(h(y)\) loses any constant solutions where \(h(y) = 0\), which should be checked separately because they are often physically important (an empty tank, a population of zero, an equilibrium). And the solution may exist only on a limited interval, for example until a tank empties.

Draining a tank: Torricelli's law

Water flowing out of a small hole at the bottom of an open tank leaves with a speed of about \(\sqrt{2gh}\), where \(h\) is the depth of water above the hole, a result known as Torricelli's law (it follows from energy conservation, as fluid mechanics courses show). If the hole has area \(a\) and a discharge coefficient \(C_d\) (about 0.6 for a sharp-edged orifice, accounting for the contraction of the jet), the volume flow rate out is \(C_d a\sqrt{2gh}\). For a tank with constant cross-sectional area \(A\), conservation of volume gives:

This equation is separable: \(h^{-1/2}\,dh = -k\,dt\) with \(k = C_d a\sqrt{2g}/A\). Integrating gives \(2\sqrt{h} = -kt + C\), so \(\sqrt{h}\) falls linearly with time:

Logistic growth

Exponential growth cannot continue forever: populations run out of food and space, an epidemic runs out of susceptible people, and the adoption of a new technology saturates the market. The logistic equation models growth that slows as it approaches a limit \(K\), the carrying capacity:

For small \(P\), the growth is nearly exponential at rate \(r\); as \(P\) approaches \(K\), growth stops. Separating and using partial fractions (Lesson 1), \(\dfrac{K}{P(K - P)} = \dfrac{1}{P} + \dfrac{1}{K - P}\), gives \(\ln\left|\dfrac{P}{K - P}\right| = rt + C\), and solving for \(P\):

The graph is an S-shaped curve whose growth is fastest at \(P = K/2\), the inflection point. Logistic curves are used to model the market share of new technologies, such as electric vehicles, the growth of microbial cultures in bioreactors, and the spread of infections in their early stages.

Other separable models

Linear first-order equations

A first-order equation is linear if it can be written in the standard form:

where \(P\) and \(Q\) depend only on \(x\). Such equations are generally not separable when \(Q \ne 0\), but they can always be solved by multiplying through by an integrating factor:

The integrating factor is chosen so that the left side becomes the derivative of a product: \(\mu y' + \mu P y = (\mu y)'\), because \(\mu' = \mu P\). The equation becomes \((\mu y)' = \mu Q\), and integrating gives:

The steps are: write the equation in standard form, compute \(\mu\) (no constant is needed in its exponent), multiply through, recognize the left side as \((\mu y)'\), integrate, and divide by \(\mu\).

Mixing problems

Mixing problems track the amount of a substance in a well-stirred tank as liquid flows in and out. The rate of change of the amount equals the rate in minus the rate out, and the rate out is the outflow rate times the current concentration. Such models describe chemical reactors, water treatment basins, the dilution of pollutants in a lake and the clearance of a drug from the blood.

If the inflow and outflow rates differ, the volume changes with time, \(V(t) = V_0 + (q_{in} - q_{out})t\), and the coefficient \(P(t) = q_{out}/V(t)\) is no longer constant, but the integrating factor method still applies.

Circuits with time-varying sources

A resistor and inductor in series, driven by a source voltage \(v(t)\), obey Kirchhoff's voltage law, \(L\dfrac{di}{dt} + Ri = v(t)\), a linear first-order equation with time constant \(\tau = L/R\). For a constant source the current rises exponentially to \(V/R\); for a sinusoidal source, the integrating factor method gives a transient that dies away and a steady sinusoidal current that lags the voltage.

Choosing a method

For a first-order equation \(y' = F(x, y)\), check in order: Is it of the form \(y' = f(x)\)? Integrate directly. Can \(F\) be factored as \(g(x)h(y)\)? Separate. Can it be written as \(y' + P(x)y = Q(x)\)? Use an integrating factor. Some equations are both separable and linear (\(y' = ky\), the cooling law); either method works. Equations that are neither are solved by special substitutions, by qualitative methods such as slope fields, or numerically with Euler's method and its more accurate successors, which MATH-300 covers.