Non-linear Differential Equation

A differential equation is an equation that relates one or more unknown functions and their derivatives. This type of equation can take on many different forms. The functions in an application typically stand in for physical quantities, the derivatives for the rates of change in those values, and the differential equation for defining the relationship between the two. Let’s start with the basics and learn what a linear differential equation is before moving on to the more complicated topic of a non-linear differential equation.

Linear Equation Differential: A differential equation that is defined by a linear polynomial in the unknown function and its derivatives is called a linear differential equation.

The general form of a linear equation is 

a₀ (x) y + a₁ (x) y’ + a₂ (x) y’’ + … + a_n (x) yⁿ = b (x)

where a₀(x), …, an(x) and b(x) are arbitrary differentiable functions that don’t have to be linear, and y′,…, y(n) are the successive derivatives of an unknown function y of the variable x.

This kind of equation is called an ordinary differential equation. If the unknown function depends on more than one variable and the derivatives in the equation are partial derivatives, then the linear differential equation can also be called a linear partial differential equation.

Nonlinear Differential Equation

A non-linear differential equation is one in which the unknown function and its derivatives don’t have a straight line when plotted in a graph (the linearity or non-linearity in the arguments of the function are not considered here). There aren’t many ways to solve nonlinear differential equations exactly, and the ones that do exist usually require the equation to have certain symmetries.

Over long periods of time, nonlinear differential equations can act in very strange ways. This is a sign of chaos. Even the most basic questions about the existence, uniqueness and extendability of solutions for nonlinear differential equations and the well-posedness of initial and boundary value problems for nonlinear partial differential equations are hard to answer, and when they are, it is considered a big step forward in mathematical theory. But if the differential equation is a correct description of a real-world physical process, then it should have a solution.

Differential equations that are linear are often used to approximate equations that are not linear. These are only close estimates that work in certain situations.

For instance, the harmonic oscillator equation is a close approximation of the nonlinear pendulum equation that works for oscillations with small amplitudes.

There are some nonlinear differential equations for which the exact solutions are known, but many of those that are essential in applications do not have known exact solutions. Occasionally, these equations can be linearized using a procedure known as expansion, in which the nonlinear elements are eliminated. This cannot be done in situations where nonlinear terms provide significant contributions to the solution; nevertheless, there are occasions when it is sufficient to keep a few “small” ones. 

General Form

A nonlinear differential equation is an equation of the form

x_n+1 = f(x_n, x_n-1,…)

where x„ is the value of x in generation n and where the recursion function f depends on nonlinear combinations of its arguments (f may involve quadratics, exponentials, reciprocals, or powers of the x„’s, and so forth). A solution is again a general formula relating x„ to the generation n and to some initially specified values, e.g., x0, x1, and so on.

Difference Between Linear and Nonlinear Equations

In the first set of examples, u is a function of x that we don’t know, while c is known constants. Ordinary differential equations and partial differential equations can be put into two large groups: linear differential equations and nonlinear differential equations. Also, differential equations can be divided into homogeneous and heterogeneous differential equations.

• The ordinary differential equation for a first-order linear system with a constant coefficient is du/dx = cu + x²
• Homogeneous second-order linear ordinary differential equation is (du²/dx²) – x(du/dx) + u = 0
• (du²/dx²) + ω²u = 0  is the ordinary differential equation for the harmonic oscillator with a homogeneous second-order linear constant coefficient.
• du/dx = u² + 4 is the first-order nonlinear heterogeneous ordinary differential equation.
• The motion of a pendulum of length L is described by the equation L (du²/dx²) + g sin u = 0

  This equation is nonlinear because of the sine function.

In the next set of examples, the unknown function u depends on two variables, x and t or x and y.

• (∂u/∂t) + t(∂u/∂x) = 0 is a homogeneous first-order linear partial differential equation.

• (∂²u/∂x²) + (∂²u/∂y²) = 0 is the Laplace equation for a homogeneous second-order linear constant coefficient elliptic type partial differential equation.

• Homogeneous third-order non-linear partial differential equation

(∂u/∂t) = 6u(∂u/∂x) − (∂³u/∂x³)

Conclusion

A nonlinear differential equation is one that is not linear with respect to the unknown function and its derivatives. Linear differential equations are frequently used to approximate nonlinear equations. These are simply approximations that work in some circumstances. The general form of a nonlinear differential equation is xn+1 = f(x_n, x_n-1,…). A harmonic oscillator equation is an approximation of the nonlinear pendulum equation that is valid for oscillations of modest amplitude and is an example of non-linear differential equations.