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Where is error in solving for constant o...

nm 11378 Maple 2024
ode dsolve differential-equations
April 21 2024
2 3

Could someone be able to spot why I get different solution when solving for the constant of integration from this Maple dsolve solution manually than when asking Maple to do it directly?

This is the ode 

ode:=x*y(x)*diff(y(x), x) = (x + 1)*(y(x) + 1);
ic:=y(1) = 1;

If I ask Maple to solve it with the IC all at once, it gives solution which odetest verifies OK.

If I ask Maple to solve it with no IC, then solve the constant myself and plug the constant back into the solution I get solution which does not verify any more.

I am not able to find why. Could someone spot the error in this? Please see worksheet below. I suspect the problem is when plugging back the constant of integration into the general solution, but have no idea now what it is. Clearly Maple did something much smarter than what I did by just plugging the constant back into the solution. May be need to specify what branch to use when plugging the constant back? but how do I know which one?
 

> 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

> 

Physics:-Version()

`The "Physics Updates" version in the MapleCloud is 1730 and is the same as the version installed in this computer, created 2024, April 17, 19:33 hours Pacific Time.`

> 

ode:=x*y(x)*diff(y(x), x) = (x + 1)*(y(x) + 1);
ic:=y(1) = 1;
sol_no_IC:=dsolve(ode);
 

x*y(x)*(diff(y(x), x)) = (x+1)*(y(x)+1)

y(1) = 1

y(x) = -LambertW(-exp(-x-1)/(x*c__1))-1

> 

sol_with_IC_direct:=dsolve([ode,ic]);
odetest(sol_with_IC_direct,[ode,ic]);

y(x) = -LambertW(-1, -2*exp(-x-1)/x)-1

[0, 0]

> 

#this verifies Maple found correct constant also:
> 

solve(rhs(sol_with_IC_direct)=rhs(sol_no_IC),c__1)

1/2

> 

#now solve for constant of integration manually. This gives invalid solution. Why?
> 

eq:= 1=limit(rhs(sol_no_IC),x=1);

1 = -LambertW(-exp(-2)/c__1)-1

> 

PDEtools:-Solve(eq,c__1);

c__1 = 1/2

> 

sol_with_ic:=eval(sol_no_IC,%)

y(x) = -LambertW(-2*exp(-x-1)/x)-1

> 

odetest(sol_with_ic,[ode,ic]);

[0, 2+LambertW(-2*exp(-2))]

 

 

Download why_wrong_solution.mw

why limit fails when having two limits a...

nm 11378 Maple 2024
limit assertion
April 19 2024
1 2

Is it wrong to call limit like this

limit(expr,[y = y0,x=x0]);

vs

limit(expr,y = y0);
limit(%,x=x0);

The first one gives internal Maple error.  Worksheet attached. I would have expected both to work the same.

> 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

> 

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1727 and is the same as the version installed in this computer, created 2024, April 17, 19:33 hours Pacific Time.`

> 

kernelopts('assertlevel'=2):
> 

expr:=3/2*(y-1)^(2/3)-3/2*x^2-c__1 = 0;
y0:=-7;
x0:=3;
limit(expr,[y = y0,x=x0]);
 

(3/2)*(y-1)^(2/3)-(3/2)*x^2-c__1 = 0

-7

3

Error, (in limit/multi/ldegree1) assertion failed

> 

expr:=3/2*(y-1)^(2/3)-3/2*x^2-c__1 = 0;
y0:=-7;
x0:=3;
limit(expr,y = y0);
limit(%,x=x0);

(3/2)*(y-1)^(2/3)-(3/2)*x^2-c__1 = 0

-7

3

-3+(3*I)*3^(1/2)-(3/2)*x^2-c__1 = 0

-33/2+(3*I)*3^(1/2)-c__1 = 0

 

 

Download limit_internal_error_maple_2024.mw

Error, (in anonymous procedure called fr...

nm 11378 Maple 2024
ode differential-equations odetest
April 19 2024
2 3

I still see these Maple internal errors in Maple 2024.

Now calling odetest.

The problem is that it is not possible to catch them.

Any suggestion what to do and what causes it?

11196

> 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

> 

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1727 and is the same as the version installed in this computer, created 2024, April 17, 19:33 hours Pacific Time.`

> 

sol:=y(x) = (exp(RootOf(-sin(x)*tanh(1/2*_Z+1/2*c__1)^2+sin(x)+exp(_Z)))+sin(x))/sin(x);
ode:=diff(y(x),x)-cot(x)*(y(x)^(1/2)-y(x)) = 0;

y(x) = (exp(RootOf(-sin(x)*tanh((1/2)*_Z+(1/2)*c__1)^2+sin(x)+exp(_Z)))+sin(x))/sin(x)

diff(y(x), x)-cot(x)*(y(x)^(1/2)-y(x)) = 0

> 

try
   odetest(sol,ode,y(x));
catch:
   print("cought error ");
end try;

Error, (in anonymous procedure called from depends) too many levels of recursion

 

 

Download too_many_levels_of_recursion_calling_odetest_maple_2024.mw

odetest does not validate correct soluti...

nm 11378 Maple 2024
ode dsolve differential-equations
April 18 2024
1 0

I think I found another clitch in odetest.

dsolve gives correct solution to this first order ode with IC. But odetest does not verify that the solution is satisfied for the IC part, but only for the ode itself. 

Below worksheet confirms the solution is also valid for the IC.   So why odetest does not give 0 for the IC part?

``

> 

restart;

19644

> 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

> 

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1727. The version installed in this computer is 1725 created 2024, April 15, 17:29 hours Pacific Time, found in the directory C:\Users\Owner\maple\toolbox\2024\Physics Updates\lib\`

> 

libname;

"C:\Users\Owner\maple\toolbox\2024\Physics Updates\lib", "C:\Program Files\Maple 2024\lib"

> 

ode:=x^2*diff(y(x),x)*cos(y(x))+1=0;
ic:=y(infinity)=Pi/3; #16/3*Pi;
sol:=dsolve([ode,ic]);
odetest(sol,[ode,ic])

x^2*(diff(y(x), x))*cos(y(x))+1 = 0

y(infinity) = (1/3)*Pi

y(x) = arcsin((1/2)*(3^(1/2)*x+2)/x)

[0, (1/3)*Pi]

> 

#we see that odetest think the solution does not verify the IC. But it does
> 

IC_eq:=Pi/3=limit(rhs(sol),x=infinity)

(1/3)*Pi = (1/3)*Pi

Download odetest_does_not_verfiy_correct_solution_maple_2024_april_18_2024.mw

I think what odetest did is not use limit when plugging in the values. That is why.

If we do not use limit, this is what happens:

IC_eq:=Pi/3=eval(rhs(sol),x=infinity)

And this explains the odetest output. It should have used limit.

ps. just in case also reported to Maplesoft support.

How to make ODESteps use the new nicer l...

nm 11378 Maple 2024
ode dsolve differential-equations
April 16 2024
1 0

I noticed that Student:-ODEs:-ODESteps does not use the newer subscripted constant of integrations for solution of odes which looks much nicer.

Is there a way to make it use same constant of integrations as dsolve() does? Compare  

This is on a worksheet using typesetting level extended. Worksheet is attached


 

> 

restart

18836

> 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

> 

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1724 and is the same as the version installed in this computer, created 2024, April 15, 17:29 hours Pacific Time.`

> 

#to make Maple use the new constant of integrations. Is this still needed in Maple 2024?
dsolve(diff(y(x),x$9)=1,arbitraryconstants=subscripted):
pdsolve(diff(psi(x,t),x$9)=0,arbitraryfunctions=subscripted):
> 

ode := diff(y(x), x$2) + 2*y(x) = 0;
Student:-ODEs:-ODESteps(ode,y(x));

ode := diff(y(x), x, x)+2*y(x) = 0

"[[,,"Let's solve"],[,,((ⅆ)^2)/(ⅆx^2) y(x)+2 y(x)=0],["•",,"Highest derivative means the order of the ODE is" 2],[,,((ⅆ)^2)/(ⅆx^2) y(x)],["•",,"Characteristic polynomial of ODE"],[,,r^2+2=0],["•",,"Use quadratic formula to solve for" r],[,,r=(0+/-([]))/2],["•",,"Roots of the characteristic polynomial"],[,,r=(-ⅈ sqrt(2),ⅈ sqrt(2))],["•",,"1st solution of the ODE"],[,,y[1](x)=cos(sqrt(2) x)],["•",,"2nd solution of the ODE"],[,,y[2](x)=sin(sqrt(2) x)],["•",,"General solution of the ODE"],[,,y(x)=C1 y[1](x)+C2 y[2](x)],["•",,"Substitute in solutions"],[,,y(x)=C1 cos(sqrt(2) x)+C2 sin(sqrt(2) x)]]"

> 

#compare to this output
dsolve(ode,y(x));

y(x) = c__1*sin(2^(1/2)*x)+c__2*cos(2^(1/2)*x)


 

Download make_step_solution_use_new_constant_of_integration.mw

 

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