# Why Is a Differential Equation Called Nonlinear?

• I
• wasi-uz-zaman
In summary: Note that an exception is made for the time variable t (the variable that we are differentiating by). We can have any crazy non-linear function of t appear in the equation, but still have an equation that is linear in x.
wasi-uz-zaman
TL;DR Summary
why do we call nonlinear differential equation as nonlinear?
hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used.
it is easy to explain nonlinearity in case of simple equation i.e when output is not proportional to the input. but how can we explain the nonlinearity of Differential equation qualitatively.
regards wasi

Linear just means that the variable in an equation appears only with a power of one. So x is linear but x2 is non-linear. Also any function like cos(x) is non-linear.

In math and physics, linear generally means "simple" and non-linear means "complicated". The theory for solving linear equations is very well developed because linear equations are simple enough to be solveable.

Non-linear equations can usually not be solved exactly and are the subject of much on-going research. Here is a brief description of how to recognize a linear equation.

Recall that the equation for a line is
y = m x + b
where m, b are constants ( m is the slope, and b is the y -intercept). In a differential equation, when the variables and their derivatives are only multiplied by constants, then the equation is linear. The variables and their derivatives must always appear as a simple first power. Here are some examples.

x'' + x = 0 is linear
x'' + 2x' + x = 0 is linear
x' + 1/x = 0 is non-linear because 1/x is not a first power
x' + x2 = 0 is non-linear because x2 is not a first power
x'' + sin(x) = 0 is non-linear because sin(x) is not a first power
x x' = 1 is non-linear because x' is not multiplied by a constant

Similar rules apply to multiple variable problems.
x' + y' = 0 is linear
x y' = 1 is non-linear because y' is not multiplied by a constant

Note, however, that an exception is made for the time variable t (the variable that we are differentiating by). We can have any crazy non-linear function of t appear in the equation, but still have an equation that is linear in x .

x'' + 2 x' + x = sin(t) is linear in x
x
' + t2x = 0 is linear in x
sin(t) x' + cos(t) x = exp(t) is linear in x

https://www.myphysicslab.com/explain/classify-diff-eq-en.html

Astronuc
wasi-uz-zaman said:
Summary:: why do we call nonlinear differential equation as nonlinear?
it is easy to explain nonlinearity in case of simple equation i.e when output is not proportional to the input.
For a linear ODE, if the input is a linear combination, then the output is a linear combination.
Consider the ODE:
$$\frac{dy}{dx} + g(x)y = f(x)$$

We can consider f(x) as the input to the system ##y'+g(x)y##, with output y(x). Any change to f leads to a change in y. Now suppose that ##f_1## leads to ##y_1## and ##f_2## leads to ##y_2##

For linear ODEs, a linear combination ##c_1f_1 + c_2f_2## leads to a linear response ##c_1y_1 + c_2y_2##:
$$\frac{d(c_1y_1+c_2y_2)}{dx}+g(x)(c_1y_1+c_2y_2) = c_1(\frac{dy_1}{dx}+g(x)y_1) + c_2(\frac{dy_2}{dx}+g(x)y_2) = c_1f_1 + c_2f_2$$

This is called the linear superposition principle and it is an important property of linear ODEs.

Perchaddition, PhDeezNutz, valenumr and 1 other person
wasi-uz-zaman said:
i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used.
To expand on what @jedishrfu wrote, an ordinary differential equation (ODE) can be represented symbolically as some combination of the independent variable and the unknown function and its derivatives like this:
##G(x, y, y', y'', \dots, y^{(n)}) = 0##
If the combination is limited to sums of constant multiples of the quantities in the parentheses above, the DE is called a linear differential equation.

Here is the (slightly) modified list that jedishrfu wrote, with additional explanation:
y'' + y = 0 is linear -- both y and y'' appear to the first power
y'' + 2y' + y = 0 is linear -- y' is multiplied by a constant
y' + 1/y = 0 is non-linear because 1/y is not a first power -- Also, the only operations allowed are addition and multiplication by a constant
y' + y2 = 0 is non-linear because y2 is not a first power
y'' + sin(y) = 0 is non-linear because sin(y) is not a first power
y y' = 1 is non-linear because y' is not multiplied by a constant

## 1. What is a nonlinear differential equation?

A nonlinear differential equation is a mathematical equation that involves both the dependent variable and its derivatives in a nonlinear way. This means that the relationship between the dependent variable and its derivatives is not a simple linear function.

## 2. How is a nonlinear differential equation different from a linear differential equation?

A linear differential equation involves only the dependent variable and its derivatives in a linear way, meaning that the relationship between them is a simple linear function. Nonlinear differential equations, on the other hand, involve more complex relationships between the dependent variable and its derivatives.

## 3. What are some examples of nonlinear differential equations?

Some examples of nonlinear differential equations include the logistic equation, the Lotka-Volterra equations, and the Navier-Stokes equations. These equations are commonly used in physics, biology, and engineering to model complex systems.

## 4. How are nonlinear differential equations solved?

Unlike linear differential equations, there is no general method for solving nonlinear differential equations. However, there are various techniques and numerical methods that can be used to approximate solutions, such as the Euler method, the Runge-Kutta method, and the shooting method.

## 5. What are the applications of nonlinear differential equations?

Nonlinear differential equations have a wide range of applications in various fields, including physics, engineering, economics, and biology. They are used to model and understand complex systems and phenomena, such as population dynamics, fluid flow, and electrical circuits.

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