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
Science Advisor
1,089
10
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
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,361
1,028
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
Science Advisor
Homework Helper
Insights Author
Gold Member
9,557
767
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
Science Advisor
1,089
10
But I still believe the operator definition holds. This agrees with your definition, Mr

LCKurtz : yourform is linear on the function y.
 
  • #7
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,557
767
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.
 

Related Threads on Linear or Nonlinear Differential Equation?

Replies
14
Views
2K
Replies
22
Views
35K
Replies
10
Views
2K
Replies
8
Views
2K
Replies
0
Views
2K
Replies
3
Views
2K
Replies
6
Views
816
Replies
3
Views
5K
Replies
8
Views
2K
Replies
2
Views
2K
Top