linear

Examples of linear in the following topics:

• Linear Equations

  • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.
  • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
  • Linear differential equations are of the form:

• Linear Approximation

  • A linear approximation is an approximation of a general function using a linear function.
  • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
  • Linear approximations are widely used to solve (or approximate solutions to) equations.
  • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
  • If one were to take an infinitesimally small step size for $a$, the linear approximation would exactly match the function.

• Nonhomogeneous Linear Equations

  • In the previous atom, we learned that a second-order linear differential equation has the form:
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • However, there is a very important property of the linear differential equation, which can be useful in finding solutions.
  • Phenomena such as heat transfer can be described using nonhomogeneous second-order linear differential equations.
  • Identify when a second-order linear differential equation can be solved analytically

• Linear and Quadratic Functions

  • Linear and quadratic functions make lines and parabola, respectively, when graphed.
  • In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
  • Linear functions may be confused with affine functions.
  • Although affine functions make lines when graphed, they do not satisfy the properties of linearity.
  • Linear functions form the basis of linear algebra.

• Second-Order Linear Equations

  • A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • Linear differential equations are of the form $Ly = f$, where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and the right hand side $f$ is a given function of the same nature as $y$ (called the source term).
  • The linear operator $L$ may be considered to be of the form:
  • The linearity condition on $L$ rules out operations such as taking the square of the derivative of $y$, but permits, for example, taking the second derivative of $y$.
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.

• Applications of Second-Order Differential Equations

  • A second-order linear differential equation can be commonly found in physics, economics, and engineering.
  • Examples of homogeneous or nonhomogeneous second-order linear differential equation can be found in many different disciplines, such as physics, economics, and engineering.
  • Therefore, we end up with a homogeneous second-order linear differential equation:
  • Identify problems that require solution of nonhomogeneous and homogeneous second-order linear differential equations

• Tangent Planes and Linear Approximations

  • The plane describing the linear approximation for a surface described by $z=f(x,y)$ is given as:

• Basic Integration Principles

  • The integral of a linear combination is the linear combination of the integrals.

• Approximate Integration

  • The function $f(x)$ (in blue) is approximated by a linear function (in red).

• Exponential Growth and Decay

  • In the long run, exponential growth of any kind will overtake linear growth of any kind as well as any polynomial growth.
  • This graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.