Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

Is it possible to integrate eq (1) in such a way that the final result will be of 1st order differential equation? 

 


 

restart

with(PDEtools)

eq := (diff(U(z), z))^3*(diff(U(z), z, z))+(diff(U(z), z))*(diff(U(z), z, z, z, z))-(diff(U(z), z, z))*(diff(U(z), z, z, z)) = 0

(diff(U(z), z))^3*(diff(diff(U(z), z), z))+(diff(U(z), z))*(diff(diff(diff(diff(U(z), z), z), z), z))-(diff(diff(U(z), z), z))*(diff(diff(diff(U(z), z), z), z)) = 0

(1)

eq1 := map(convert, eq, diff); eq2 := map(int, lhs(eq1), z)-C1 = 0

(diff(U(z), z))^3*(diff(diff(U(z), z), z))+(diff(U(z), z))*(diff(diff(diff(diff(U(z), z), z), z), z))-(diff(diff(U(z), z), z))*(diff(diff(diff(U(z), z), z), z)) = 0

 

(1/4)*(diff(U(z), z))^4-(diff(diff(U(z), z), z))^2+(diff(diff(diff(U(z), z), z), z))*(diff(U(z), z))-C1 = 0

(2)

``


 

Download inttegration.mw

The wikipedia website below contains a general description of Doyle spirals but not the full mathematics of their construction.  

https://en.wikipedia.org/wiki/Doyle_spiral

The website below apparently contains the html coding for an animated display of Doyle spirals, but I am not familiar with this coding language.

https://bl.ocks.org/robinhouston/6096950

Can anyone direct me to 1) the complete mathematics describing the construction of a Doyle spiral and/or

                                        2) a Maple worksheet which codes for the display of a Doyle spiral?

I have created some plots of inverse primes  like this example1.pdf .

The filled color-shape in the middle is what I want do do with all areas in this picture or in other pictures.

In other words the goal is to fill the differnt areas in the print with different colors.

So I need to find the points of the Polygons, as I have done by hand with that yellow Polygon.

A procdure that is ready will give give the crosspoints of the lines.

These are the line-coordinates (the 1st number ist the number of iterations)

2*L[1]=number of lines in L

L:=[14, [[1, 1], [1, 26]], [[1, 26], [26, 26]], [[26, 26], [26, 37]], [[26, 37], [37, 37]], [[37, 37], [37, 39]], [[37, 39], [39, 39]], [[39, 39], [39, 20]], [[39, 20], [20, 20]], [[20, 20], [20, 23]], [[20, 23], [23, 23]], [[23, 23], [23, 30]], [[23, 30], [30, 30]], [[30, 30], [30, 70]], [[30, 70], [70, 70]], [[70, 70], [70, 45]], [[70, 45], [45, 45]], [[45, 45], [45, 34]], [[45, 34], [34, 34]], [[34, 34], [34, 32]], [[34, 32], [32, 32]], [[32, 32], [32, 51]], [[32, 51], [51, 51]], [[51, 51], [51, 48]], [[51, 48], [48, 48]], [[48, 48], [48, 41]], [[48, 41], [41, 41]], [[41, 41], [41, 1]], [[41, 1], [1, 1]]]

These are the crosspoints:

cp := [[23, 26], [30, 37], [32, 37], [26, 30], [48, 45], [39, 34], [41, 34], [45, 41]]

To plot the pdf I used this code:

poly2 := [[32, 32], [34, 32], [34, 34], [39, 34], [39, 39], [37, 39], [37, 37], [32, 37]]

poly 2 is just an axample, how it looks like when its ready.

display(seq(line(op(L[i])), i = 2 .. 2*L[1] + 1), polygonplot([poly2], color = "Resene GoldenTainoi", axes = none, style = polygon), color = blue, thickness = 0.8);

So I hope, you can help me :)

This is a beautyfull way to paint a prime  by just printing the remainders of the recursive dividing of the inverse prime in lines.

Thanks a lot,

Arno

An important contribution to the integrability of non-linear differential equation, made by the E.S.Cheb.Terrab, is the Abel Inverse Riccati(AIR) equation. Recently I get an idea of make generalization of the equation using some technique introduced in Kamke's books. I write it in latex. I hope you guys can give me some advice. I have deleted some questions I asked before because I think it is more appropriate to create a post to share my ideas with you.Final_version.pdf

For example my friend emailed me.
His email address is oneman@gmail.com, the email content is HELLO.
How does Maple print HELLO to the screen so I can see it?
Thanks for your help!

I make new cone puzzle. However, I cat't make function l(θ). Can maple solve this puzzle?

θ=90 degree is a YouTube problem I found.