Linear or Nonlinear Differential Equation?

[physicsforum7]

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

a_n(x)y⁽ⁿ⁾ + a_n-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.

 

[Bacle2]

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).

 

[SammyS]

WolframAlpha characterizes it as a first-order linear ordinary differential equation.

 

[physicsforum7]

Where on Wolfram Alpha are you able to find something that tells you whether a DE is linear or nonlinear?

 

[LCKurtz]

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]

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

LCKurtz : yourform is linear on the function y.

 

[LCKurtz]

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.

 

[physicsforum7]

My professor gave the "can be put in linear form" definition; hence my confusion. Thank you guys for easing my mind.