# 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.

## Answers and Replies

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

LCKurtz : yourform is linear on the function y.

LCKurtz
Science Advisor
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.

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.

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