Determine the order of differentiation for this partial differential eqn

In summary, the conversation discusses the equivalence between two notations for partial derivatives and the concept of linear and non-linear differential equations. The participants also mention using observation to identify linear equations and provide examples.
  • #1
karush
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Homework Statement
determine the order of the given partial differential equation;also state whether the equation is linear or nonlinear
Relevant Equations
The ordinary differential equation is said to be linear if F is a linear function of the variables y,y,...,y(n); a similar definition applies to partial differential equations.
The order of a differential equation is the order of the highest derivative that appears in the equation.
Screenshot 2022-08-31 11.03.05 AM.png

ok I posted this a few years ago but replies said there was multiplication in it so I think its a mater of format
##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{xy}##

textbook
 
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  • #2
here is an example I found from online ... but its probably beyond what my OP is asking for
Screenshot 2022-08-31 11.23.54 AM.png
 
  • #3
For "Relevant Equations", you should not just put "definitions". You should put your exact definition of the order of a PDE.
 
  • #5
Greg Bernhardt said:
Also use double pound ## for inline math
double pound is not very collaborate with other latex editors this is the only forum I know of that asks that
 
  • #6
ok it looks these are just observation problems but ? about linear non linear pare
 
  • #7
karush said:
double pound is not very collaborate with other latex editors this is the only forum I know of that asks that
But you're here, and that's the way it works here...

"When in Rome, ..."
 
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  • #8
karush said:
##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{xy}##
If memory serves, the two notations are not the same; i.e., the order in which partials are taken is different.
The above should read "##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{yx}##", not ##u_{xy}## as you wrote.
In the first notation, you first take the partial with respect to y, and then take the partial of that with respect to x. In the second notation, you take the partial with respect to the first variable listed, and then with the second.
 
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  • #9
It's not clear to me what we are being asked to do. The first two posts are not about the same issue regarding PDEs. And there is no attempt to answer either.
 
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  • #10
Mark44 said:
If memory serves, the two notations are not the same; i.e., the order in which partials are taken is different.
The above should read "##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{yx}##", not ##u_{xy}## as you wrote.
In the first notation, you first take the partial with respect to y, and then take the partial of that with respect to x. In the second notation, you take the partial with respect to the first variable listed, and then with the second.
IIRC, the two are barely-ever different from each other.
 
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  • #11
FactChecker said:
It's not clear to me what we are being asked to do. The first two posts are not about the same issue regarding PDEs. And there is no attempt to answer either.
it basically to solve just by observation like 2 order and linear no calculation is needed
 
  • #12
karush said:
it basically to solve just by observation like 2 order and linear no calculation is needed
In your first post, what do you think the answers are? We can comment on that.
 
  • #13
WWGD said:
IIRC, the two are barely-ever different from each other.
By Schwarz's theorem they are the same, providing both partial derivatives are [edit]continuous differentiable[/edit].
 
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  • #14
FactChecker said:
In your first post, what do you think the answers are? We can comment on that.
ok I would like that I don't know how to discern linear for non Linear from observation
the yellow highlighted problems are ones I have done but want to do the 15-24

the link to the whole Boyce textbook in the OP if need.

BTW I am retired so just doing this on my own might audit de at UHM next year

Mahalo ahead

Screenshot 2022-09-03 12.35.14 PM.png
 
  • #15
A linear differential equation is one where all you have are derivatives being multiplied by functions of the underlying variable and being added together##x^2y'' + \cos(x)y +2=0## is linear.
##\sin(y')=y## is not because you are doing something to a derivative other than adding and multiplying by defined functions of x.
##u_x u_y=0## is not because you are doing something to a derivative other than adding them and multiplying by defined functions of x and y
 
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FAQ: Determine the order of differentiation for this partial differential eqn

1. What is the purpose of determining the order of differentiation for a partial differential equation?

The order of differentiation for a partial differential equation helps to classify the equation and determine the appropriate methods for solving it. It also provides insight into the complexity of the equation and the number of independent variables involved.

2. How is the order of differentiation determined for a partial differential equation?

The order of differentiation is determined by counting the number of times the highest order derivative appears in the equation. This is known as the order of the equation.

3. What are the different orders of differentiation for partial differential equations?

The orders of differentiation for partial differential equations can range from first order to higher orders such as second, third, and so on. The order is determined by the highest derivative present in the equation.

4. Can the order of differentiation change during the solution process?

No, the order of differentiation remains constant throughout the solution process. However, the order of the equation can change if additional terms or variables are introduced.

5. Why is it important to correctly determine the order of differentiation for a partial differential equation?

Correctly determining the order of differentiation is crucial for selecting the appropriate method to solve the equation. It also helps to avoid errors and ensures an accurate solution to the problem at hand.

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