Why Is a Differential Equation Called Nonlinear?

[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

 

[jedishrfu]

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

quoted from this article

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

 

[bigfooted]

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.

 

[Mark44]

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