Linear or Nonlinear Differential Equation?

1. Aug 31, 2012

physicsforum7

1. The problem statement, all variables and given/known data

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

2. Relevant equations

1. The problem statement, all variables and given/known data[/b]

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

2. Relevant equations

$\sqrt{xy'+2x2}$=5

3. 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 31, 2012

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).

3. Aug 31, 2012

SammyS

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

4. Aug 31, 2012

physicsforum7

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

5. Aug 31, 2012

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.

6. Aug 31, 2012

Bacle2

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

LCKurtz : yourform is linear on the function y.

7. Aug 31, 2012

LCKurtz

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