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linear and non-linear ODE

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PG1995

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Hi

Please have a look on the attachment. You can find my questions there. Please help me with them. Thanks a lot.

The equation dy/dx = y/(y+x) is non-linear because it cannot be written in the form shown in green highlight in the attachment.

ODE: An ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable. This means there could be more than one dependent variables such as y, z, w, etc, but they will all depend on one independent variable, say, x.

Regards
PG
 

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Last edited:
Hi,

Im not sure if this helps, but a linear function is a function where multiplication of the variable by a constant results in a multiplication of the function by that same constant.

Consider the function y=f(x):

y=f(x)
A*y=f(A*x)

The function is linear because multiplication of x by the constant A resulted in multiplication of y by the same constant A. OR alternately vice versa.

Example1:
y=3*x
A*y=3*x*A ???
Yes, so it is linear.

Example2:
y=x^2
We only have to disprove one case to show it is not linear, if it is not.
Set x=3, then y=9.
Set x=2*3, then y=36.
Did multiplication of x by 2 result in multiplication of y by 2 ?
The answer is no, so it's not linear.


Example3:
y=sin(x)
A*y=sin(A*x) ???
If this is not true we only have to disprove one case.
y=sin(x)
with x=pi/2 we get
y=1
With x=2*pi/2 we get
y=0
So multiplication of x by 2 resulted in multiplication of y by zero (0), so sin(x) is not linear.

More generally for any function f(x):
If:
y1=f(x)
and:
y2=f(x*A)
then:
if y2/y1=A then f(x) is linear, else it is not linear.


Someone once said, a linear differential equation is one that you can solve, a nonlinear one you can't :)
Often nonlinear differential equations require numerical solution.
 
Last edited:
PG1995,

ODE: An ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable. This means there could be more than one dependent variables such as y, z, w, etc, but they will all depend on one independent variable, say, x.

That's wrong. You can have many independent variables per equation, but only one dependent variable. How could it be otherwise? To be linear, the dependent variable or its derivative cannot show up more than once in a term, and must not be trancendental, such as e^y or sin(y).

In your example, there is a y*dy term, so it is not linear with respect to y. However, by rearranging the example to y*dx/dy-x = y , we can see that it is linear with respect to x, when x is considered the dependent variable.

Ratch
 
PG,

Nowhere in that reference does it say that more than one dependent variable can exist.

Ratch
 
Hi Ratch

That's why I said it is based on Wikipedia definition. Some person told me that there can more than one dependent variable but independent is to be only one. You said "You can have many independent variables per equation, but only one dependent variable". Am I missing something here? The Wikipedia also says that there can be only one independent variable but according to you there can be many independent variables. Please help me to clarify it. Thanks.

Regards
PG
 
PG,

Some person told me that there can more than one dependent variable but independent is to be only one.

Well, think about it. How would you define an equation with more than one dependent variable?

You said "You can have many independent variables per equation, but only one dependent variable". Am I missing something here?

I will give you an example. A topographic map. Z = F(X) + G(Y). Z is the dependent variable and X and Y are the two independent variables.

The Wikipedia also says that there can be only one independent variable but according to you there can be many independent variables. Please help me to clarify it.

For an ODE, ordinary differentials like dy/dx appear. For equations with more than one independent variable, partial differentials appear like δZ/δx and δZ/δy . Then you have a partial differential equation (PDE).

Ratch
 
Hi,

Not sure what you guys are arguing here, but consider:

x=t^2+2*t+1
y=3*t+2

For this parametric set, x and y depend on t.

Set of ODE's with one dependent variable t:
dy/dt=1-a*y(t)-b*z(t)
dz/dt=-c*y(t)-d*z(t)

r^2=x^2+y^2
We could say that x and y are independent variables.
 
Last edited:
MrAl,

x=t^2+2*t+1
y=3*t+2

For this parametric set, x and y depend on t.

Set of ODE's with one dependent variable t:
dy/dt=1-a*y(t)-b*z(t)
dz/dt=-c*y(t)-d*z(t)

I was thinking of a relationship described by only one equation, not a set of equations. In any case, dy/dt and dz/dt indicates that 't' is the independent variable.

r^2=x^2+y^2
We could say that x and y are independent variables.

Indeed so.

Ratch
 
Hi,

Well ODE's often come in sets, so that's why i didnt really understand the question too well because there it is apparent there is one independent variable. And parametric equations are valid equations too so i cant just arbitrarily dismiss them :)
 
Last edited:
MrAl,

Well ODE's often come in sets, so that's why i didnt really understand the question too well because there it is apparent there is one independent variable. And parametric equations are valid equations too so i cant just arbitrarily dismiss them :)

Although sets of equations exist, we were only talking about one equation. No one is dismissing parametric equations, we just were not discussing them.

Ratch
 
PG,



Well, think about it. How would you define an equation with more than one dependent variable?



I will give you an example. A topographic map. Z = F(X) + G(Y). Z is the dependent variable and X and Y are the two independent variables.



For an ODE, ordinary differentials like dy/dx appear. For equations with more than one independent variable, partial differentials appear like δZ/δx and δZ/δy . Then you have a partial differential equation (PDE).

Ratch

Thank you, Ratch.

So, you were talking about PDE. I had ODE in mind. So, it's clear that ODE needs to have one independent variable. But I believe I was wrong when I said that there could be more than one dependent variable in ODE (perhaps, I didn't exactly understood that person who was telling me about this).

Regards
PG
 
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