non-linear differential equation

(noun)

nonlinear partial differential equation is partial differential equation with nonlinear terms

Related Terms

Examples of non-linear differential equation in the following topics:

  • 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:
    • A wave function which satisfies the non-relativistic Schrödinger equation with $V=0$.

  • Predator-Prey Systems

    • The relationship between predators and their prey can be modeled by a set of differential equations.
    • 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.

  • 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:
    • The equation describes the self-limiting growth of a biological population.
    • Letting $P$ represent population size ($N$ is often used instead in ecology) and $t$ represent time, this model is formalized by the following differential equation:
    • In the equation, the early, unimpeded growth rate is modeled by the first term $rP$.

  • 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

  • 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:

  • 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).
    • where $D$ is the differential operator $\frac{d}{dt}$ (i.e.
    • When $f(t)=0$, the equations are called homogeneous second-order linear differential 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.

  • Linear Approximation

    • A linear approximation is an approximation of a general function using a linear function.
    • Linear approximations are widely used to solve (or approximate solutions to) equations.
    • Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem states that:
    • Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of $f$ at $(a, f(a))$.
    • Since the line tangent to the graph is given by the derivative, differentiation is useful for finding the linear approximation.

  • Linear Equations in Standard Form

    • A linear equation written in standard form makes it easy to calculate the zero, or $x$-intercept, of the equation.
    • Standard form is another way of arranging a linear equation.
    • In the standard form, a linear equation is written as:
    • However, the zero of the equation is not immediately obvious when the linear equation is in this form.
    • Convert linear equations to standard form and explain why it is useful to do so

  • 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

  • The Equation of a Line

    • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
    • This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
    • 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.
    • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.