Linear or Nonlinear Differential Equation?

In summary, the conversation discusses whether a given differential equation, specifically ##\sqrt{xy'+2x^2}=5##, is linear or nonlinear. The definition of a linear DE is also debated, with some sources stating that it must be in the form ##a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... a_0(x)y=g(x)## while others saying any DE that can be put in that form is linear. Ultimately, it is determined that the given DE is indeed linear.
  • #1

Homework Statement



Is this differential equation linear or nonlinear? Assume that y' means dy/dx.

Homework Equations



1. Homework Statement [/b]

Is this differential equation linear or nonlinear? Assume that y' means dy/dx.

Homework Equations



[itex]\sqrt{xy'+2x2}[/itex]=5

The Attempt at a Solution



It seems to me that squaring both sides and subtracting 2x2 from both sides renders a linear equation:

xy' + 2x2=25.

xy' = 25 - 2x2

This is of the form

an(x)y(n) + an-1(x)y(n-1) + ... a0(x)y=g(x)

if you take a0 = 0... isn't it?

The problem is that my professor says otherwise; namely, that it is a nonlinear DE.

Where am I making my mistake?

Thanks.
 
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  • #2
The way I know it, in order to decide if the operator is linear,you express it (the

differential operator) in the form f(y) and then see if f is linear in that sense, i.e.,

if h is another function, is f(y+h)=f(y)+f(h) ,is f(cy)=cf(y).
 
  • #3
WolframAlpha characterizes it as a first-order linear ordinary differential equation.
 
  • #4
Where on Wolfram Alpha are you able to find something that tells you whether a DE is linear or nonlinear?
 
  • #5
I don't think you will find unanimity on this question. Some sources say a linear DE is one that has the form ##a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... a_0(x)y=g(x)## while others any DE that can be put in that form is linear. It depends on the definition your professor gave.
 
  • #6
But I still believe the operator definition holds. This agrees with your definition, Mr

LCKurtz : yourform is linear on the function y.
 
  • #7
LCKurtz said:
I don't think you will find unanimity on this question. Some sources say a linear DE is one that has the form ##a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... a_0(x)y=g(x)## while others any DE that can be put in that form is linear. It depends on the definition your professor gave.

Bacle2 said:
But I still believe the operator definition holds. This agrees with your definition, Mr

LCKurtz : yourform is linear on the function y.

But the question was about this DE:

##\sqrt{xy'+2x^2}=5##

Certainly the left side is not linear in y' as it stands, although it can be put in the linear form by squaring.
 
  • #8
My professor gave the "can be put in linear form" definition; hence my confusion. Thank you guys for easing my mind.
 

What is the difference between a linear and nonlinear differential equation?

A linear differential equation is an equation where the dependent variable and its derivatives appear in a linear form, meaning there are no exponents or products. In contrast, a nonlinear differential equation is an equation where the dependent variable and its derivatives appear in a nonlinear form, meaning there are exponents and/or products.

How do you solve a linear differential equation?

A linear differential equation can be solved by using various methods such as separation of variables, integrating factors, or the method of undetermined coefficients. These methods involve manipulating the equation to isolate the dependent variable and integrating it to find the general solution.

What techniques can be used to solve a nonlinear differential equation?

There is no one specific technique that can be used to solve all types of nonlinear differential equations. Depending on the form of the equation, techniques such as substitution, transformation, or numerical methods may be used to obtain a solution.

Why are nonlinear differential equations important in science?

Nonlinear differential equations are important in science because they can accurately model real-world systems that exhibit nonlinear behavior. This allows scientists to make predictions and understand complex systems in a more realistic manner. Nonlinear equations are also useful for studying phenomena such as chaos and turbulence.

What are some applications of linear and nonlinear differential equations?

Linear differential equations are commonly used in physics, engineering, and economics to model systems that behave in a linear manner. Nonlinear differential equations are often used to model biological systems, chemical reactions, and weather patterns. They are also used in fields such as finance, ecology, and population dynamics.

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