Ordinary differential equations describe how an unknown quantity changes with respect to one independent variable. They provide a mathematical language for modeling motion, population change, chemical reactions, and many other evolving processes.
A differential equation relates an unknown function to one or more of its derivatives. An ODE is a differential equation in which the unknown function depends on only one independent variable. For example, if y=f(x), then x is the independent variable and y is the dependent variable.
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What is an ordinary differential equation?
An equation involving an unknown function of one independent variable and one or more derivatives of that function.
What does the order of an ODE mean?
It is the order of the highest derivative appearing in the equation.
What is the difference between an independent and a dependent variable?
The independent variable is the input variable; the dependent variable is the unknown quantity whose value varies with that input.
Why is Newton's law of motion an example of an ODE?
Position depends on time, and the law relates position to its second time derivative, acceleration.
Name two fields in which ODEs are used.
Examples include physics, astronomy, chemistry, biology, ecology, meteorology, and economics.
Quiz questions
Which expression is the explicit form of y′′+y=x?Answer: y′′=x−y
Which statement correctly classifies the equation y′+xy=0?Answer: It is linear, homogeneous, and has a variable coefficient.
In an explicit system with unknowns y1(x),…,ym(x), what does the system's dimension m count?Answer: The number of component functions in the system.
For the initial value problem y′=F(x,y), which condition is sufficient for a local existence-and-uniqueness guarantee near (x0,y0)?Answer: F and its partial derivative with respect to y are continuous near (x0,y0).
What does Sturm–Liouville theory study in an important class of second-order linear ODE problems?Answer: Boundary-value problems involving eigenvalues and eigenfunctions.