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Linear or Nonlinear Differential Equation?

  • #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.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Bacle2
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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
SammyS
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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
LCKurtz
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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
Bacle2
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But I still believe the operator definition holds. This agrees with your definition, Mr

LCKurtz : yourform is linear on the function y.
 
  • #7
LCKurtz
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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.
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.
 

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