# Linear or Nonlinear Differential Equation?

## 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

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

## The Attempt at a Solution

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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
Staff Emeritus
Homework Helper
Gold Member
WolframAlpha characterizes it as a first-order linear ordinary differential equation.

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

LCKurtz
Homework Helper
Gold Member
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
Homework Helper
Gold Member
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.
$\sqrt{xy'+2x^2}=5$