linear equation

(noun)

a polynomial equation of the first degree (such as $x = 2y - 7$)

Related Terms

  • simultaneous equations
  • differential equation

Examples of linear equation 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 parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:
    • 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:
  • 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.
    • Linear differential equations are differential equations that have solutions which can be added together to form other 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
  • 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).
    • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
    • (Otherwise, the equations are called nonhomogeneous equations.)
    • A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.
  • 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.
    • The equation of motion is given as:
    • 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
  • Predator-Prey Systems

    • The predator–prey equations are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey.
    • As differential equations are used, the solution is deterministic and continuous.
    • However, a linearization of the equations yields a solution similar to simple harmonic motion with the population of predators following that of prey by 90 degrees.
    • The solutions to the equations are periodic.
    • Identify type of the equations used to model the predator-prey systems
  • Series Solutions

    • The power series method is used to seek a power series solution to certain differential equations.
    • The power series method is used to seek a power series solution to certain differential equations.
    • In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
    • Let us look at the case know as Hermit differential equation:
    • Using power series, a linear differential equation of a general form may be solved.
  • 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.
    • Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of $f$ at $(a, f(a))$.
  • Linear and Quadratic Functions

    • 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.
    • Linear functions form the basis of linear algebra.
    • If the quadratic function is set equal to zero, then the result is a quadratic equation.
    • The solutions to the equation are called the roots of the equation.
  • Separable Equations

    • Separable differential equations are equations wherein the variables can be separated.
    • Non-linear differential equations come in many forms.
    • A separable equation is a differential equation of the following form:
    • The original equation is separable if this differential equation can be expressed as:
    • Integrating such an equation yields:
  • Logistic Equations and Population Grown

    • A logistic equation is a differential equation which can be used to model population growth.
    • The logistic function is the solution of the following simple first-order non-linear differential equation:
    • More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative $t$, which slows to linear growth of slope $\frac{1}{4}$ near $t = 0$, then approaches $y = 1$ with an exponentially decaying gap.
    • The equation describes the self-limiting growth of a biological population.
    • In the equation, the early, unimpeded growth rate is modeled by the first term $rP$.
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