@nm 

Thanks

You are right for this type of ODE its solved by integrating and rather easy to solve. 

@vv 

Thanks

Yes, indeed its not function with multiplicative inverse., but a inverse function see 

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

@nm 

Thanks

Just annother last question :how do you get the blue answers between the gray boxes ?  

@acer 

Thanks

Yes, i tried that first with the <pre> tags( no <quote> tags) ( made in past some websites , so tags were familiar) 
But get not the text between the two tags.
There were two  <P>  lines and the blocks get separated 
Tried several times and gave up too soon.

@nm 

DEtools['odeadvisor']( y(x) = 1*ln(cos(diff(y(x), x)))+(diff(y(x), x))*tan(diff(y(x), x)), y(x), 'help' );
                          [_dAlembert]
----------------------------------
DEtools['odeadvisor']( y(x) = ln(cos(diff(y(x), x)))+(diff(y(x), x))*tan(diff(y(x), x)), y(x), 'help' );
                          [_dAlembert]

For x= 1 you get a dAlembert ODE

@Carl Love 

Thanks

Now i can make a gray box  ( distracted by the two A's )

@nm 

Thanks

I overlooked this icon.

@nm 

In this form how can the x be vanished ? 

y = x*f(p) + g(p)   

Only if  f(p) = 0 , because x . 0 = 0 
y = 0 + g(p) 
y = g(y')   ( p = dy/dx)  is this still a d'Alembert ( Maple say yes ..no quadrature) 
and g is function

For x= 1 you get  y = 1. f(p)+g(p)  d'Alembert ODE
For x= 0 you get a quadrature ODE

 

@vv 

Well, at least with the odeadvisor you know if you are dealing with a standard form or variant of a particular ODE  

@Preben Alsholm 

Thanks!

Works good now.
As you instructed.  
 

Thanks

What i do sometimes for learning a new program is looking how the F1..  keys are assigned in menu or in general.
In maple menu its hidden : its under <edit>split or join that F3 and F4 are explained + Shift F3....

That's not handy done in Maple !.. i like t see in the menu right away the F3 and F4 etc 
Also i like to see in Maple in a save folder ( same as in Microsoft Windows 10)  that i can copy en save,delete worksheets.
And also naming worksheets inthe save folder of Maple : same as in Microsoft Windows 10)
That were some sidenotes and maybe it will be chanced in Maple in the near future. 

I split the regions and the de command is working, that's good, but a error message still exists : Error, unexpected single forward quote. .some text is causes this 

@janhardo 

Indeed, i was not sure if the answer from Rouben was correct, but i saw today that geometry has also his interest.
I took over  the example from dr John Val and assume that his work is correct done ( but that was not the case)

@tomleslie 

Thanks

That's a ingenious way to look at this with the use of similar triangles
Scaling of the coordinates that's the idea  
It was a excercise meant for highschool students to learn vector calculations, but its solved on a quicker way.

@tomleslie 

Thanks

Well done, looks not so complicated with the geom3d package , compared with the original calculation!
Checking this is a little bit more complicated , because its done with standard vector calculation :  direction vector ,normalvector ,etc

You could do this calculalton on the same way as in the example with maple too, but tha needs more insicht in vector calculations.
and how to perform this in maple

Author is http://www.johnval.nl/school/wiskunde/  see linear algebra pdf ( end of it)

Must again look to the reasoning of Rouben:   The ratio of the areas of the rectangles EFGH and SJIK equals (ML/NL)^2.
This remains true even if you replace the rectangle EFGH by any convex shape.
   
 

@Rouben Rostamian  

Thanks

As i studied the example it is 76 +2/25  as answer for the the green area.

It was dr. John Val who did the math , so i did not check his answer