Determine the order of differentiation for this partial differential eqn

[karush]

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.

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

 

[karush]

here is an example I found from online ... but its probably beyond what my OP is asking for

 

[FactChecker]

For "Relevant Equations", you should not just put "definitions". You should put your exact definition of the order of a PDE.

 

[Greg Bernhardt]

Also use double pound ## for inline math

 

[karush]

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

 

[karush]

ok it looks these are just observation problems but ? about linear non linear pare

 

[Mark44]

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, ..."

 

[Mark44]

$\dfrac{\partial u^2}{\partial x\partial y}$ 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.

 

[FactChecker]

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.

 

[WWGD]

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.

 

[karush]

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

 

[FactChecker]

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.

 

[pbuk]

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].

 

[karush]

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

Spoiler: exercises page

 

[Office_Shredder]

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