Differential equation

A differential equation is an equation whose unknown is a function, and which relates that function to its derivatives. An ordinary equation such as $x^2=9$ is solved by numbers; a differential equation such as $\dot{y}=2y$ is solved by functions $y(t)$.

General form

For an unknown function $y$ of one independent variable $x$, the general form is

$$F\bigl(x,y,y',y'',\ldots,y^{(n)}\bigr)=0$$

where $y'$, $y''$, … are the successive derivatives of $y$.

where $y'$, $y''$, … are the successive derivatives of $y$ and $F$ is a given expression of its $n+2$ arguments. The order of the equation is the order $n$ of the highest derivative that appears. Newton's second law of motion for a body of mass $m$ at position $x(t)$,

$$m\frac{d^{2}\vec r}{dt^{2}}=\vec F$$

is of order 2: the unknown $x$ appears together with its second derivative, while the net force $F$ is given.

Classification

A differential equation is linear when the unknown function and its derivatives appear only to the first power, are never multiplied together, and are never the argument of another function. A linear equation of order $n$ reads

$$a_n(x)\,y^{(n)}+a_{n-1}(x)\,y^{(n-1)}+\cdots+a_1(x)\,y'+a_0(x)\,y=g(x)$$

with coefficients $a_{0}$, …, $a_{n}$ that depend on the independent variable $x$ only, and a given source term $g(x)$;

When $g(x)=0$ the equation is homogeneous.

with coefficients $a_{0}$, …, $a_{n}$ that depend on the independent variable only, and a given source term $g(x)$; when $g=0$ the equation is homogeneous:

$$ \begin{cases} \dot{y}=2y & \text{linear, homogeneous} \\ y'=y^{2} & \text{nonlinear: square of the unknown} \\ \ddot{\theta}+\dfrac{g}{L}\sin\theta=0 & \text{nonlinear: sine of the unknown} \end{cases} $$

If the unknown function depends on several independent variables, only partial derivatives appear, and the equation is a partial differential equation. The heat equation

$$\frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}}$$

for the temperature $u(x,t)$ of a one-dimensional bar is the standard example. An equation whose unknown depends on a single independent variable is an ordinary differential equation, treated in Ordinary differential equation; partial differential equations are treated in Partial differential equation.

Solutions

A solution of a differential equation is a function that satisfies it: substituting the function and its derivatives into the equation must give an identity in the independent variable, on some interval. For example $y=Ce^{2t}$ solves $\dot{y}=2y$ for every constant $C$, since

$$y=Ce^{2t} \ \Longrightarrow\ \dot{y}=2Ce^{2t}=2y$$

Every value of $C$ gives a solution, so the equation has infinitely many. The whole family is its general solution; for a linear equation of order $n$ the general solution contains exactly $n$ arbitrary constants. Prescribing values of the solution and of its derivatives fixes those constants and selects one particular solution.

In the simplest case the derivative is a given function of $x$ alone. Integrating both sides of $y'=x$ then gives the general solution,

$$y'=x \ \Longrightarrow\ \int y'\,dx=\int x\,dx \ \Longrightarrow\ y=\frac{x^{2}}{2}+C$$

and the single condition $y(0)=3$ fixes the constant:

$$3=\frac{0^{2}}{2}+C \ \Longrightarrow\ C=3 \ \Longrightarrow\ y=\frac{x^{2}}{2}+3$$

A condition imposed at one point is an initial condition, and an equation together with initial conditions is an initial value problem:

Conditions imposed at two different points are boundary conditions, and the problem is a boundary value problem. Most differential equations are satisfied by none of the elementary functions, and then no formula for the solution exists.

Methods of solution

No formula solves all differential equations, just as no formula solves all algebraic equations. Each class has its own method:

• Ordinary differential equations: a single independent variable; the method depends on the order. First-order equations are solved by separation of variables, by an integrating factor, or by the exactness criterion; linear equations with constant coefficients, of the second and higher orders, by their characteristic equation.
• Partial differential equations: several independent variables; solved by separation of variables or, at the first order, by the method of characteristics.
• Numerical methods: Euler's method and its higher-order refinements step along the direction field, when only numbers are required.
• Qualitative methods: equilibria, stability, and long-term behaviour, read directly from the equation.
• Series and transforms: the unknown written as a power series, or the equation turned into an algebraic one by the Laplace transform.

Where differential equations arise

Most laws of nature state how a quantity changes, so they are written as differential equations: mechanics states Newton's second law as $m\ddot{\mathbf{r}}=\mathbf{F}$, the decay of a radioactive sample and the growth of an unchecked population follow $\dot{y}=ky$, and the conduction of heat through a solid is the partial differential equation $u_{t}=\alpha u_{xx}$. Solving the equation turns the law into a prediction, such as the quantity of the sample remaining at a later time or the temperature at a point of the solid.

See also

• Ordinary differential equation
• Partial differential equation

Further reading

• Differential equation, Wikipedia
• Differential Equation, Wolfram MathWorld
• 18.03SC Differential Equations, MIT OpenCourseWare