Difference between linear and nonlinear Differential Equation|Linear verses nonlinear DE

In this video, we will explore the difference between linear and nonlinear differential equations. Differential equations are mathematical equations that describe the behavior of dynamic systems. They are used in various fields such as physics, engineering, economics, and biology.
A differential equation is said to be linear if it can be written in the form:
a_n(x)y^(n)(x) + a_(n-1)(x)y^(n-1)(x) + ... + a_1(x)y'(x) + a_0(x)y(x) = f(x)
where y(x) is the dependent variable, x is the independent variable, y' is the first derivative of y with respect to x, y'' is the second derivative of y with respect to x, and so on. The coefficients a_n(x), a_(n-1)(x), ..., a_1(x), a_0(x) are functions of x, and f(x) is a function of x that represents the forcing function.
On the other hand, a differential equation is said to be nonlinear if it cannot be written in the above form. In other words, if the equation contains products or powers of the dependent variable and its derivatives, it is considered to be nonlinear. Nonlinear differential equations are more complex and difficult to solve than linear differential equations.
One important feature of linear differential equations is that the superposition principle holds. This means that if y1(x) and y2(x) are solutions to a linear differential equation, then any linear combination of y1(x) and y2(x), such as c1y1(x) + c2y2(x), is also a solution. This property allows us to find general solutions to linear differential equations by finding a set of solutions and then combining them in a linear combination.
Another important feature of linear differential equations is that they have constant coefficients. This means that the coefficients a_n, a_(n-1), ..., a_1, a_0 are constants and do not depend on x. Constant coefficient linear differential equations have well-known solutions, such as exponential functions and sinusoidal functions.
Nonlinear differential equations, on the other hand, do not have the superposition principle or constant coefficients. This means that finding general solutions to nonlinear differential equations is much more difficult. In many cases, it is not possible to find explicit solutions, and numerical methods or approximation techniques are used instead.
Nonlinear differential equations can exhibit a wide range of behaviors, including chaos and bifurcations. Chaos is a phenomenon where small changes in the initial conditions lead to vastly different outcomes, and it can be observed in some nonlinear differential equations. Bifurcations are sudden changes in the behavior of the system as a parameter is varied. Nonlinear differential equations can exhibit a rich and complex range of behaviors that are not seen in linear differential equations.
In conclusion, the difference between linear and nonlinear differential equations lies in their form and properties. Linear differential equations are simpler and have constant coefficients, while nonlinear differential equations are more complex and do not have these properties. Linear differential equations have the superposition principle, which allows for easy calculation of solutions, while nonlinear differential equations can exhibit complex and unexpected behaviors. Understanding the difference between these two types of equations is important in many areas of science and engineering.

[@ib9653]

The DE at 2:24 is a linear differential equation. It satisfies all the conditions you gave in the beginning

[@hearthacker5565]

Bro here dependent variable is "x" not "y". And x has exponent 2 which implies this is non linear. May Allah bless you

[@jayeshdave680]

@IB Did you find the correct answer to it..i m agree with you

[@Johan-or1wj]

Yes, this is a linear D.E, and there is a mistake in this video, because we should judge on D.E depending on the linearity of the "Dependent" variable (y) and its derivatives , not on the "Independent" variable (x).

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what about sin(dy/dx)

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Sir give some examples of LDE...

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[@musfidurrahman8406]

sir..if the degree of the highest order is 1/2 ..will it be linear then?

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[@gunjanrathore9337]

d2y/dx2 + x2. dy/dx + 4xy = x sinx
Is it question linear or non linear

[@emonhossain8556]

dv/ds + dv/dt = v
Is this equation linear or non linear and why?

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[@karangujarati4589]

But what, when (y^2)(d2y/dx2) i.e. square of dependant variable is in multiple with second order differential term?! Is it also consider as non linera!?

[@davel7037]

Yes, non linear

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[@tusharbehera8956]

Sir if there is a term like sin(y) and y is the dependent variable then would it be considered non linear?

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Watch video of khan academy on difderential equations for this

[@RahulKumar-wd1yf]

Sir , dy / dx = y square . Is this linear or Non - linear Differential equation ?
Please reply and give reason also

[@NumberX]

Nonlinear, because exponent of dependent variable is 2. Please watch @02:31. Thanks.

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Is xy' - yx' = c linear or nonlinear?

[@BayAreaChill]

That's non-linear.

[@pradnyajagtap3638]

If exponent of dependent variable is 0 and other comments is 1 then is it linear?

[@NumberX]

Yes it is. The exponent of dependent variable must not be greater than 1. It means it can either 0 or 1.
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Hlo sir how many integrating factor does a non exact differential equation have ?????
Plz reply me sir

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what about if there e^x, is it linear or nonlinear

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[@NumberX]

Thanks for being in touch with us. Here degree of D.E. equals 1, exponent of dependent variable equals 1. No terms contain product of y and dy/dx. Hence, linear.

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