> with(DEtools,Dchangevar): > de := 'diff'(sin(x)*diff(f(x),x),x)/sin(x) ; > collect(simplify(Dchangevar({x = arccos(y), f(x)=F(y)}, de, x,y)),diff) ;
Maple R4 answers:
-y*diff(F(y),y) + (1-y^2)*diff(F(y),y$2) ;
I expected:
-2*y*diff(F(y),y)+ (1-y^2)*diff(F(y),y$2) ;
In the online help about Dchangevar, the first example in "Examples/Additional examples" gives:
> de := diff(y(x),x$2) = y(x)*diff(y(x),x)/x ; > Dchangevar({x=exp(t),y(x)=Y(t)},de,x,t) ;
Maple answers, after some simplication:
diff(Y(t),t$2) = Y(t)*diff(Y(t),t) ;
Answer expected:
diff(Y(t),t$2) = (Y(t)+1)*diff(Y(t),t) ;
It seems that Dchangevar mistakes if the order of the differential equation is >=2 and if the
change of variable is not linear.
Is there a bug in Maple or in my mind?
The bug is removed in Maple V Release 5. (U. Klein)
As an alternative, you can use PDEtools[dchange], which is able to perform non-linear
changes of variables and also in PDEs. In your example,
> de := 'diff'(sin(x)*diff(f(x),x),x)/sin(x) ; > tr := {x = arccos(y), f(x)=F(y)}; tr := {x = arccos(y), f(x) = F(y)} > normal(dchange(tr,de)); / 2 \ / 2 \ |d | 2 /d \ |d | -|--- F(y)| y - 2 |-- F(y)| y + |--- F(y)| | 2 | \dy / | 2 | \dy / \dy /
PDEtools is a package distributed in the Share Library of Maple R4. The last version of PDEtools is found at
http://lie.uwaterloo.ca/pdetools.html (Canada)
or
http://dft.if.uerj.br/pdetools.html (Brazil)
I get the same as you do! There must be a bug in Dchangevar.
I checked the variable transformation with a procedure of my own ( vsubs ) and could confirm the supposed results of G. Thomas and also reproduce the wrong answers that Dchangevar gave :
> vsubs([y],[x=arccos(y)],[f(x)=F(y)],diff(sin(x)*diff(f(x),x),x)/sin(x)); / | | 2 1/2 /d \ |-cos(arccos(y)) (1 - y ) |-- F(y)| - | \dy / \ //d \ \ ||-- F(y)| y / 2 \| 2 1/2 |\dy / 2 1/2 |d || sin(arccos(y)) (1 - y ) |----------- - (1 - y ) |--- F(y)|| | 2 1/2 | 2 || \(1 - y ) \dy // \ | | |/sin(arccos(y)) | / > simplify(%); / 2 \ / 2 \ |d | 2 /d \ |d | -|--- F(y)| y - 2 |-- F(y)| y + |--- F(y)| | 2 | \dy / | 2 | \dy / \dy / > with(DEtools,Dchangevar); [Dchangevar] > Dchangevar({x=arccos(y),f(x)=F(y)},diff(sin(x)*diff(f(x),x),x)/sin(x),x,y); / | 2 1/2 /d \ |-cos(arccos(y)) (1 - y ) |-- F(y)| | \dy / \ / 2 \\ 2 |d || + sin(arccos(y)) (1 - y ) |--- F(y)||/sin(arccos(y)) | 2 || \dy // > simplify(%); / 2 \ / 2 \ /d \ |d | |d | 2 -|-- F(y)| y + |--- F(y)| - |--- F(y)| y \dy / | 2 | | 2 | \dy / \dy / > vsubs([t],[x=exp(t)],[y(x)=Y(t)],diff(y(x),x,x)-y(x)*diff(y(x),x)/x); / / 2 \\ | /d \ |d || exp(-t) |-exp(-t) |-- Y(t)| + exp(-t) |--- Y(t)|| | \dt / | 2 || \ \dt // /d \ Y(t) exp(-t) |-- Y(t)| \dt / - ---------------------- exp(t) > simplify(%); / 2 \ /d \ |d | -exp(-2 t) |-- Y(t)| + exp(-2 t) |--- Y(t)| \dt / | 2 | \dt / /d \ - Y(t) |-- Y(t)| exp(-2 t) \dt / > Dchangevar({x=exp(t),y(x)=Y(t)},diff(y(x),x,x)-y(x)*diff(y(x),x)/x,x,t); > 2 d --- Y(t) /d \ 2 Y(t) |-- Y(t)| dt \dt / -------- - -------------- 2 2 exp(t) exp(t)
I have noticed the bug in DEtools a long time ago. I had sent Maple a worksheet summarizing where the bug occurs, but never received any follow up.
If I recall, what I had observed was that DEtools did not know how to apply the chain rule properly.
I discovered this while trying to get maple to perform the same set of calculations I had done on paper. Had the change of variable worked properly, I would have reduced my ODE to one which has the hypergeometric function as a solution. Otherwise, Maple could not solve it. I was trying to reproduce some work I had found in a paper I was using for my research.