Linear vs Nonlinear Differential Equations
An equation containing at least one differential coefficient or derivative of an unknown variable is known as a differential equation. A differential equation can be either linear or non-linear. The scope of this article is to explain what is linear differential equation, what is nonlinear differential equation, and what is the difference between linear and nonlinear differential equations.
Since the development of calculus in the 18th century by the mathematicians like Newton and Leibnitz, differential equation has played an important role in the story of mathematics. Differential equations are of great importance in mathematics because of their range of applications. Differential equations are at the heart of every model we develop to explain any scenario or event in the world whether it is in physics, engineering, chemistry, statistics, financial analysis, or biology (the list is endless). In fact, until calculus became an established theory, proper mathematical tools were unavailable to analyze the interesting problems in the nature.
Resulting equations from a specific application of calculus may be very complex and sometimes not solvable. However, there are ones that we can solve, but may look alike and confusing. Therefore, for easier identification differential equations are categorized by their mathematical behaviour. Linear and nonlinear is one such categorization. It is important to identify the difference between linear and nonlinear differential equations.
What is a Linear Differential Equation?
Suppose that f: X→Y and f(x)=y, a differential equation without nonlinear terms of the unknown function y and its derivatives is known as a linear differential equation.
It imposes the condition that y cannot have higher index terms such as y2, y3,… and multiples of derivatives such as
It also cannot contain non linear terms such as Sin y, ey^-2, or ln y. It takes the form,
where y and g are functions of x. The equation is a differential equation of order n, which is the index of the highest order derivative.
In a linear differential equation, the differential operator is a linear operator and the solutions form a vector space. As a result of the linear nature of the solution set, a linear combination of the solutions is also a solution to the differential equation. That is, if y1 and y2 are solutions of the differential equation, then C1 y1+ C2 y2 is also a solution.
The linearity of the equation is only one parameter of the classification, and it can further be categorized into homogenous or non-homogenous and ordinary or partial differential equations. If the function is g=0 then the equation is a linear homogeneous differential equation. If f is a function of two or more independent variables (f: X,T→Y) and f(x,t)=y , then the equation is a linear partial differential equation.
Solution method for the differential equation is dependent on the type and the coefficients of the differential equation. The easiest case arises when the coefficients are constant. Classic example for this case is Newton’s second law of motion and its various applications. Newton’s second law produces a second order linear differential equation with constant coefficients.
What is a Nonlinear Differential Equation?
Equations that contain nonlinear terms are known as non-linear differential equations.
All above are nonlinear differential equations. Nonlinear differential equations are difficult to solve, therefore, close study is required to obtain a correct solution. In case of partial differential equations, most of the equations have no general solution. Therefore, each equation has to be treated independently.
Navier-Stokes equation and Euler’s equation in fluid dynamics, Einstein’s field equations of general relativity are well known nonlinear partial differential equations. Sometimes the application of Lagrange equation to a variable system may result in a system of nonlinear partial differential equations.
What is the difference between Linear and Nonlinear Differential Equations?
• A differential equation, which has only the linear terms of the unknown or dependent variable and its derivatives, is known as a linear differential equation. It has no term with the dependent variable of index higher than 1 and do not contain any multiple of its derivatives. It cannot have nonlinear functions such as trigonometric functions, exponential function, and logarithmic functions with respect to the dependent variable. Any differential equation that contains above mentioned terms is a nonlinear differential equation.
• Solutions of linear differential equations create vector space and the differential operator also is a linear operator in vector space.
• Solutions of linear differential equations are relatively easier and general solutions exist. For nonlinear equations, in most cases, the general solution does not exist and the solution may be problem specific. This makes the solution much more difficult than the linear equations.