## Ode exact solutions...

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How to find the exact solution of ODE's

## can we express 'X' in terms of 'L'?...

can we express 'X' in terms of 'L'? i.e., X = (some const)*L XintoL.mw

## evalb() evaluates addition theorem to "false"...

Hello,
can anyone tell me why the following expression evaluates to false in Maple?

restart;
evalb(sin(x + y) = sin(x)*cos(y) + cos(x)*sin(y));
false

The underlying term "sin(x+y) = sin(x) cos(y) + cos(x) sin(y)" is one of the addition theorems that have been proven to be valid.

Best regards
Jan

## How can I solve a system of ODE in rkf45 method wi...

I tried to solve a ODE system using rkf45 with shooting technique. But I have a lot of errors. How to resolve this...

See the attachment: Shoot_Blasius.mw

## Substituting out Radicals and (a*x..........) typ...

I have some long expressions that would be more readable if common sections were substituted out. There are many sets of radicals often stacked inside each other.  The other one is (a*x..........) I see repeated in some other expressions. indets is good, but is there a way to use it to select  (a*x..........) types
At these eexpressions are returned from procdures would probably put them in an array/table with their substitution components.
I dont need to substitute everything. What I have done below is reasonable for reading and seeing the structure.

What would be a good approact here?

 > restart
 >
 (1)
 > vals:=[a=18,b=21,c=7,d=-29,e=-37,f=-56]
 (2)
 > vars[1]:=x:vars[2]:=y:
 > eqn:= 8*(2*((a^2-2*a*c+b^2+c^2)*(4*a*c*f-a*e^2-b^2*f+b*d*e-c*d^2)^2)^(1                 /2)+(-8*a*f+2*d^2)*c^2+(8*a^2*f-2*a*d^2+2*a*e^2+2*b^2*f-2*b*d*e)*c-2*a^2*e^2-2*                 a*b^2*f+2*a*b*d*e)^(1/2)*(a*c-1/4*b^2)/(4*a*c-b^2)^2*(vars[1]-(1/4*b*e-1/2*c*d)/(a*c-\                 1/4*b^2))-8*csgn((4*a*c*f-a*e^2-b^2*f+b*d*e-c*d^2)*(Complex(1)*a+Complex(-1)*c-                 b))*(a*c-1/4*b^2)*(2*((a^2-2*a*c+b^2+c^2)*(4*a*c*f-a*e^2-b^2*f+b*d*e-c*d^2)^2)^                 (1/2)+(8*a*f-2*d^2)*c^2+(-8*a^2*f+2*a*d^2-2*a*e^2-2*b^2*f+2*b*d*e)*c+2*a^2*e^2+                 2*a*b^2*f-2*a*b*d*e)^(1/2)/(4*a*c-b^2)^2*(vars[2]-(-1/2*a*e+1/4*b*d)/(a*c-1/4*b^2))
 (3)
 > length(eqn)
 (4)
 > simplify(eval(eqn,vals))
 (5)
 > indets(eqn)
 (6)
 > Subs:=[((a^2 - 2*a*c + b^2 + c^2)*(4*a*c*f - a*e^2 - b^2*f + b*d*e - c*d^2)^2)=A^2,          (-8*a*f + 2*d^2)*c^2 + (8*a^2*f - 2*a*d^2 + 2*a*e^2 + 2*b^2*f - 2*b*d*e)*c - 2*a^2*e^2 - 2*a*b^2*f + 2*a*b*d*e=B^2,            f*b^3 - b^2*d*e - (-(-4*c*f + e^2)*a - c*d^2)*b=C]:
 > eqn1:=simplify(eqn,Subs)
 (7)
 > Subsnumeric:=eval(Subs,vals)
 (8)
 > simplify(eval(eqn1,[(rhs=lhs)~(Subsnumeric)[],vals[] ]))
 (9)

## Solve keeps running without returning the results...

I was trying to solve a system of eight cubic equations, with eight variables. Note that the particular solutions should exist and my goal is to find all  possible solutions using solve. However, when executing, the solve keeps running for a whole day and did not throw any results. I also set the parameter infolevel[solve] to be 3 and find out that it is stuck in the step "GroebnerBasis: computing a factored plex basis using Groebner[Solve]". Can anyone tell me how to deal with that? Here's the Maple file.

solve_test.mw

## Section plane solids...

This is still a  starting procedure and let's see what can be added?

 > restart; # Define the procedure to draw a cylinder along the x-axis and a specifically positioned plane CylinderAndPlane := proc(r, h, alpha_deg, beta_deg, P, axis_length)     local alpha, beta, cylinder, plane, pointPlot, display, nx, ny, nz, px, py, pz, annotations, plane_type, titleStr, grafiek;  # Added: titleStr     uses plots, LinearAlgebra;       # Convert angles from degrees to radians     alpha := alpha_deg * Pi / 180;     beta := beta_deg * Pi / 180;     # Determine the normal vector based on angles     nx := cos(alpha) * sin(beta);     ny := sin(alpha) * sin(beta);     nz := cos(beta);     # Point P is directly used as given coordinates     px, py, pz := op(P);     # Cylinder along the x-axis     cylinder := plots:-implicitplot3d(y^2 + z^2 = r^2, x = 0 .. h, y = -r .. r, z = -r .. r, style = surface, color = "LightBlue", transparency = 0.5);     # Determine the type of plane based on angles alpha and beta     if beta_deg = 90 then         plane_type := "yz";         plane := plots:-implicitplot3d(x = px, x = px - 10 .. px + 10, y = -axis_length .. axis_length, z = -axis_length .. axis_length, style = surface, color = "Yellow", transparency = 0.5);     elif alpha_deg = 90 and beta_deg = 0 then         plane_type := "xz";         plane := plots:-implicitplot3d(y = py, x = -axis_length .. axis_length, y = py - 10 .. py + 10, z = -axis_length .. axis_length, style = surface, color = "Green", transparency = 0.5);     elif beta_deg = 0 then         plane_type := "xy";         plane := plots:-implicitplot3d(z = pz, x = -axis_length .. axis_length, y = -axis_length .. axis_length, z = pz - 10 .. pz + 10, style = surface, color = "Blue", transparency = 0.5);     else         plane_type := "arbitrary";         plane := plots:-implicitplot3d(nx * (x - px) + ny * (y - py) + nz * (z - pz) = 0, x = -axis_length .. axis_length, y = -axis_length .. axis_length, z = -axis_length .. axis_length, style = surface, color =            "Red", transparency = 0.7);     end if;     # Mark point P     pointPlot := plots:-pointplot3d([px, py, pz], symbol = solidcircle, symbolsize = 10, color = "Red");     # Create dynamic title - New     titleStr := cat("Plane: ", plane_type, "\nAlpha: ", sprintf("%.2f", alpha_deg), " deg\nBeta: ", sprintf("%.2f", beta_deg), " deg\nPoint: [", sprintf("%.2f", P[1]), ", ", sprintf("%.2f", P[2]), ", ", sprintf("%.2f", P[3]), "]");     # Display everything together - Modified: titleStr added in the display function     grafiek := plots:-display(cylinder, plane, pointPlot, axes = normal, scaling = constrained, labels = ["x", "y", "z"], title = titleStr);     return grafiek; end proc: # Example call to the procedure with coordinates of P and setting the axis length # Alpha and Beta are now angles in degrees, P is a list of coordinates, axis_length is the length of the coordinate axes CylinderAndPlane(15, 50, 0, 90, [15, 5, 5], 30);  # For yz-plane #CylinderAndPlane(5, 15, 90, 0, [5, 5, 5], 10);  # For xz-plane #CylinderAndPlane(5, 15, 0, 0, [5, 5, 5], 10);   # For xy-plane #CylinderAndPlane(5, 55, 45, 45, [5, 5, 5], 10); # For arbitrary plane

## How do I calculate and plot an Orthogonal Trajecto...

How do I calculate and plot an Orthogonal Trajectory on Maple 2024?

Here's the equation of the contour lines:

x*y^2 - x^2 - y^2 = k

I need to make it pass through a precise point on my contour lines graph, and everything I do doesn't seem to work.

## Trace: What is an unknow procedure and is it traca...

I have tried to understand why ODESteps cannot find a solution for cases where an ODE can separated into a polynomial expression and an ODE.

or example, when I trace like this

restart;
trace(Student:-ODEs:-ODESteps:-ModuleApply);
printlevel:=10;
Student:-ODEs:-ODESteps:-ModuleApply(x*(diff(y(x), x)) = 0);
printlevel:=1


trace is tracing an unknown procedure

and not showing any further details of this procedure.
What unknow function/procedure is traced?

Can, in this particular case, the message "Cannot compute integral"   be traced (it’s not an error message, which are traceable).

I have tried to increase printlevel which reveals some unkown procedures but did not get down to this one. Hence my questions.

## how to get the coefficient of x in this expression...

sol := y = -3283/4253 - (3283*x)/4253, How can I determine the value of the coefficient of x?
How can I take the value of the coefficient of x? Thank you.

## How can I convert a list containing three points a...

At here https://mathematica.stackexchange.com/questions/297104/how-can-i-convert-a-list-containing-three-points-and-its-equation-passing-three is discussing about "How can I convert a list containing three points and its equation passing three points to text file?" With mylist

mylist := [[[-12, 2, -1], [-11, 1, -5], [-10, -2, 3], 10*x + 6*y + z + 109 = 0], [[-12, 2, -1], [-11, 1, -5], [-10, 6, 3], 2*x - 2*y + z + 29 = 0], [[-12, 2, -1], [-11, 1, -5], [-9, 5, -7], 3*x - y + z + 39 = 0]]

How can I get the LaTeX file like this?

\documentclass[12pt,a4paper]{article}
\usepackage[letterpaper,margin=1.2in]{geometry}
\usepackage{enumitem}
\begin{document}
\begin{enumerate}[label=\arabic*)]
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$,\quad $C(-10; -2; 3)$,\quad $(P) : 10 x+6 y+z+109=0$
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$,\quad $C(-10; 6; 3)$,\quad $(P) : 2 x-2 y+z+29=0$
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$,\quad $C(-9; 5; -7)$,\quad $(P) : 3 x-y+z+39=0$
\end{enumerate}
\end{document}

## why Maple gives extra solutions to this basic ode?...

Given the ode   y''(x)*y'(x)=0, clearly it has solutions for y''=0 and y'=0.

These are y=c1+x*c2 and y=c1. But Maple gives 3 solutions

 > interface(version);

 > ode:=diff(y(x),x$2)*diff(y(x),x)=0;  > dsolve(ode)  > dsolve(diff(y(x),x$2)=0)

 > dsolve(diff(y(x),x)=0)

Where did the third solution come from? The 3 solutions are correct ofcourse, but why 3? There should only be two.

FYI, I am using

Physics:-Version()

The "Physics Updates" version in the MapleCloud is 1746. The

version installed in this computer is 1745 created 2024, May

I have to check earlier Maple versions to see if same thing happens there.

## why ODESteps(ode) unable to solve this first order...

This ode can be solved by just looking at it

ode:=(x+y(x))*diff(y(x),x)=0;

We see the solution is y=-x and y=c__1 because either (x+y)=0 or y'=0

But for some reason ODESteps(ode) says it cannot compute integral.

Any idea why?

 > interface(version);

 > ode:=(x+y(x))*diff(y(x),x)=0; Student:-ODEs:-ODESteps(ode);

 >

update:

Here is another simpler example that also confused it

 > interface(version);

 > ode:=x*diff(y(x),x)=0; Student:-ODEs:-ODESteps(ode);

 >
 >

update

Here is another one which it gets wrong.

 > interface(version);

 > ode:=y(x)*diff(y(x),x)=0; Student:-ODEs:-ODESteps(ode);

 > dsolve(ode);

 >

The correct solution is given by dsolve, which is y=0 and y=constant (I do not know why dsolve put minus sign in front of the constant, but it is still correct).

## Why does this command only works in Maple and not ...

I am stuck this command works seemlessly in Maple:

ThermophysicalData:-CoolProp:-Property(D, T = 20*Unit('degC'), P = 760*Unit('mmHg'), water)

but it does not work in Maple Flow. Does anyone knows why? Thank you so much for your help in the matter.