linear 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 $rP$.