Differential Equations and Their Language, differential equations lesson

Water flows into a tank through one pipe and out through another. At any moment you can read the water level and how fast it is changing, but neither number by itself tells you the level ten minutes from now. What the physics supplies is a rule for the rate of change, and the level over time is a function that has to obey that rule. This lesson sets up the words used to describe such equations, from ordinary and partial to order, linearity and the two meanings of homogeneous, and shows how to check that a proposed function really is a solution.

Key terms

Differential equation

Definition A differential equation is an equation that relates an unknown function to one or more of its derivatives, and possibly to the independent variable and to known functions. The unknown function is the function the equation is solved for, so the answer is a whole function rather than a number. A solution on an interval $I$ is a function that has every derivative appearing in the equation at each point of $I$ and makes the equation true at every point of $I$ .

In plain words The equation is a rule that the unknown function and its rates of change must obey everywhere on an interval. Solving it means finding functions that follow the rule, and checking a candidate means substituting it into the equation.

Example Take the equation $y' = -2y$ for an unknown function $y(t)$ , where $y' = dy/dt$ . The candidate $y(t) = 3e^{-2t}$ has derivative $-6e^{-2t}$ by the chain rule, and $-2y$ is also $-6e^{-2t}$ , so

y' = -6e^{-2t} = -2y

holds for every real $t$ . The same function has $y(0) = 3$ . As a model of decay it would be used for $t \ge 0$ , but as a mathematical solution it works on the whole real line. The candidate $y = 3 + t$ fails. Its derivative is $1$ , while $-2y = -6 - 2t$ , and the two agree only at $t = -\tfrac{7}{2}$ . Agreement at a single point is not agreement on an interval, so $3 + t$ is not a solution.

Ordinary and partial differential equations

Definition An ordinary differential equation, or ODE, is a differential equation whose unknown function or functions depend on a single independent variable, so only ordinary derivatives such as $dy/dt$ and $d^2y/dt^2$ appear. A partial differential equation, or PDE, involves an unknown function of two or more independent variables together with some of its partial derivatives. For a function $u(x, t)$ , the partial derivative $\partial u/\partial t$ is the rate of change in $t$ while $x$ is held fixed, and $\partial u/\partial x$ is the rate of change in $x$ while $t$ is held fixed.

In plain words Count the inputs of the unknown function. One input, often time, gives an ODE. Several inputs, such as position and time, give a PDE, and each partial derivative says which input moves while the others stay put.

Example A falling object has height $y(t)$ at time $t$ , measured upward, and $y' = dy/dt$ . Its velocity is $y'$ and its acceleration is $y''$ , so ignoring air resistance it obeys the ODE $y'' = -g$ , where $g$ is the gravitational acceleration. The temperature $u(x, t)$ at position $x$ along a thin rod at time $t$ depends on two inputs, and a standard model for it is the heat equation

\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2}

with a constant diffusivity $\kappa > 0$ . It is a PDE, and both of its derivatives come from the same function $u$ .

Order of an equation

Definition The order of a differential equation is the order of the highest derivative of the unknown function that appears in it. For a PDE it is the highest total order of the partial derivatives that appear. The power to which a derivative is raised has no effect on the order, so the order of an equation is a different thing from the degree of a polynomial.

In plain words Look for the derivative taken the most times. How often the unknown has been differentiated sets the order, while squares or cubes of terms say nothing about it.

Example For $y(t)$ with $y' = dy/dt$ , the equation $(y')^2 + y^3 = t$ is first order, even though it contains a square and a cube. The equation $y'' + 3y' + 2y = 0$ is second order, even though every term appears only to the first power. The heat equation above is second order, because $\partial^2 u/\partial x^2$ appears. An equation of order $n$ usually needs $n$ pieces of starting data to single out one solution, such as a starting position and a starting velocity for a second-order equation of motion.

Dependent and independent variables

Definition In an equation for an unknown function $y(t)$ , the input $t$ is the independent variable and the output $y$ is the dependent variable, since its value depends on the chosen $t$ . A parameter is a constant that specifies the model, such as a mass, a damping coefficient or a cross-sectional area. It stays fixed while $t$ varies, though it may differ from one version of the model to another. Primes always mean derivatives with respect to the independent variable, so that variable has to be named, as in $y' = dy/dt$ or $y' = dy/dx$ .

In plain words The independent variable is the input you choose, often time. The dependent variable is the quantity you track. Parameters are the fixed settings of the model.

Example Return to the tank. Let $y(t)$ be the water level in meters at time $t$ in minutes, let $Q_{\text{in}}$ and $Q_{\text{out}}$ be the volume flow rates in cubic meters per minute, and let the tank have a constant cross-sectional area $A$ in square meters. A change in volume spread over the area $A$ changes the level by that volume divided by $A$ , so

\frac{dy}{dt} = \frac{Q_{\text{in}} - Q_{\text{out}}}{A}.

Both sides are measured in meters per minute. Here $t$ is the independent variable, $y$ is the dependent variable and $A$ is a parameter. If the outflow through a drain at the bottom depends on the level, for example $Q_{\text{out}} = k\sqrt{y}$ with a constant $k > 0$ , the equation still has the single dependent variable $y$ , but it becomes nonlinear.

Linear and nonlinear equations

Definition An ODE of order $n$ for $y(t)$ is linear if it can be written as

\sum_{k=0}^{n} a_k(t)\, y^{(k)} = g(t),

that is, $a_n(t)y^{(n)} + \cdots + a_1(t)y' + a_0(t)y = g(t)$ , where $y^{(0)} = y$ , the coefficients $a_k$ and the right side $g$ are given functions of $t$ alone, and $a_n$ is not the zero function. An equation that cannot be written this way is nonlinear. Coefficients may vary with $t$ , but $y$ and its derivatives may not appear to powers other than the first, in products with each other or inside other functions. A PDE is linear when it has the same form in $u$ and its partial derivatives, with coefficients that depend only on the independent variables.

In plain words In a linear equation the unknown and its derivatives enter in the simplest possible way, each multiplied by a known function of the independent variable and then added up. Linearity is a property of the equation, not of the graphs of its solutions.

Example The equations $y'' + 3y' + 2y = 0$ and $y' + (\cos t)\,y = 0$ are linear, and so is the heat equation. The cosine in the second one is a known coefficient, because it depends only on $t$ . The equations $y' = \cos y$ , $y' + y^2 = 0$ and $yy' = t$ are nonlinear, because the unknown sits inside a cosine, is squared, or multiplies its own derivative. Linear does not mean that the solution graphs are straight lines. The curve $y = e^{-t}$ solves the linear equation $y' + y = 0$ , since $y' = -e^{-t} = -y$ .

Homogeneous and nonhomogeneous linear equations

Definition A linear equation as above is homogeneous if its right side $g(t)$ is zero for every $t$ in the interval, and nonhomogeneous if $g$ is not the zero function there. The function $g$ is called the forcing term or input. For a homogeneous linear equation the zero function is a solution, and if $y_1$ and $y_2$ are solutions, then so is $c_1y_1 + c_2y_2$ for all constants $c_1$ and $c_2$ . For a nonhomogeneous one the zero function is not a solution, and the difference of two solutions solves the homogeneous equation with the same left side.

A second, unrelated meaning applies to first-order equations $y' = f(x, y)$ with $y' = dy/dx$ . Such an equation is called homogeneous when its right side depends only on the ratio $y/x$ , so that $y' = F(y/x)$ for $x \ne 0$ . An equation of this kind can be nonlinear.

In plain words For a linear equation, homogeneous means that nothing drives the system from outside. In the ratio sense, it means the slope stays the same when $x$ and $y$ are scaled by the same nonzero factor. When you meet the word, ask which of the two structures is meant.

Example The equation $y' + 2y = 0$ is homogeneous. Its solutions $Ce^{-2t}$ include the zero function, and sums and multiples of them have the same form. The equation $y' + 2y = \sin t$ is nonhomogeneous, and the zero function fails it, because $\sin t$ does not vanish on any interval. The two meanings can disagree. The equation

y' = 1 + \frac{y}{x}, \qquad x \ne 0,

is homogeneous in the ratio sense, with $F(v) = 1 + v$ . Written as the linear equation $y' - y/x = 1$ , it is nonhomogeneous, because its forcing term is $1$ . The equation $y' = (y/x)^2$ is homogeneous in the ratio sense and nonlinear.

Check yourself

Classify the equation $y''' + t^2y = \cos t$ for $y(t)$ and the equation $u_t = u_{xx} + u^2$ for $u(x, t)$ , where $u_t = \partial u/\partial t$ and $u_{xx} = \partial^2 u/\partial x^2$ . For each, say whether it is ordinary or partial, give its order, and say whether it is linear.

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The first is an ODE, because $y$ depends only on $t$ . Its highest derivative is $y'''$ , so it is third order. It is linear, with coefficients $1$ and $t^2$ and forcing term $\cos t$ , and since the forcing term is not zero it is nonhomogeneous.

The second is a PDE, because $u$ depends on $x$ and $t$ and partial derivatives appear. Its highest derivative $u_{xx}$ is second order, and the term $u^2$ makes it nonlinear.

A classmate says that $(y')^2 = 4y$ is second order because of the square, and that $y' + (\sin t)\,y = 0$ is nonlinear because it contains a sine. Here $y' = dy/dt$ . Correct both claims.

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The highest derivative in $(y')^2 = 4y$ is $y'$ , so the equation is first order. The square does not change the order. It makes the equation nonlinear, which is a separate property.

In $y' + (\sin t)\,y = 0$ the sine is a coefficient that depends only on $t$ , and $y$ and $y'$ appear to the first power, so the equation is linear, first order and homogeneous. By contrast, the equation $y' + \sin y = 0$ is nonlinear, because there the unknown is inside the sine.

Check that $y_1 = 2$ and $y_2 = 2 + e^{-2t}$ both solve $y' + 2y = 4$ . Is $y_1 + y_2$ a solution? Which equation does $y_2 - y_1$ solve?

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For $y_1 = 2$ the derivative is $0$ , and $0 + 2 \cdot 2 = 4$ . For $y_2$ the derivative is $-2e^{-2t}$ , and adding $2y_2 = 4 + 2e^{-2t}$ gives $4$ . Both are solutions on the whole real line.

The sum $y = 4 + e^{-2t}$ gives $y' + 2y = -2e^{-2t} + 8 + 2e^{-2t} = 8$ , which is not $4$ , so the sum is not a solution. Each solution already produces the forcing term $4$ , so together they produce $8$ . That is the practical meaning of nonhomogeneous.

The difference $y_2 - y_1 = e^{-2t}$ gives $y' + 2y = -2e^{-2t} + 2e^{-2t} = 0$ , so it solves the homogeneous equation $y' + 2y = 0$ . For that equation sums of solutions do work. If $z_1$ and $z_2$ solve it, then $(z_1 + z_2)' + 2(z_1 + z_2)$ splits into $z_1' + 2z_1$ plus $z_2' + 2z_2$ , which is $0 + 0$ .

Differential Equations and Their Language

Video 1

Key terms