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deTaylor - fhub - 05-22-2015 01:24 PM
Here's a small CAS function deTaylor for the HP-Prime (similiar to deSolve) which I've originally written for Xcas. It approximately solves 1st- and 2nd-order differential equations (with initial conditions) as n-th degree Taylor polynomial. 1st order y'=f(x,y) with y(x0)=y0: deTaylor(f,[x,y],[x0,y0],n) 2nd order y''=f(x,y,y') with y(x0)=y0 and y'(x0)=y0': deTaylor(f,[x,y,z],[x0,y0,z0],n) (I'm using z instead of y', because y' can't be used as input) Here's the function definition: (you can directly copy&paste it into the Prime-emulator commandline in CAS-mode) Code:
Example 1: y'=x*y^2+1 with y(0)=1 (6th-degree approximation): deTaylor(x*y^2+1,[x,y],[0,1],6) Example 2: y''=x*y*y' with y(1)=2 and y'(1)=3 (5th degree): deTaylor(x*y*z,[x,y,z],[1,2,3],5) Maybe it's useful for someone, Franz RE: deTaylor - salvomic - 05-22-2015 06:10 PM
thank you! please help to control with an differential equation still not solvable in Prime (but now ok in XCas): y'=(x+y)^2 Is it correct to input deTaylor((x+y)^2, [x,y], [0,0], 7) ? I get (17/315)x^7+(2/15)x^5+(⅓)x^3 The general solution of equation (XCas) is TAN(x-c)-x With Taylor I've taylor(TAN(x)-x), x, 6) = (⅓)x^3+(2/15)x^5+x^7+o(x) It should be ok, isn't it? Salvo RE: deTaylor - fhub - 05-22-2015 08:27 PM
(05-22-2015 06:10 PM)salvomic Wrote: thank you!Not exactly, you should enter order 7 (not 6), then you get the same result as with deTaylor (despite of the error term "x^8*order_size(x)"): taylor(tan(x)-x), x=0, 7) Franz RE: deTaylor - salvomic - 05-22-2015 08:31 PM
(05-22-2015 08:27 PM)fhub Wrote: Not exactly, you should enter order 7 (not 6), then you get the same result as with deTaylor (despite of the error term "x^8*order_size(x)"): right! thank you. Parisse has already added the solution for y'=(x+y)^2 in XCas, but it was after the last FW was ready. I hope this will be added in the next firmware. In the meantime your deTaylor is helpful also for this equation ;-) Salvo RE: deTaylor - fhub - 05-22-2015 09:44 PM
(05-22-2015 08:31 PM)salvomic Wrote: Parisse has already added the solution for y'=(x+y)^2 in XCas, but it was after the last FW was ready. I hope this will be added in the next firmware.Well, in the meantime you could also use the following function: Code:
Works only for ODEs of the type y'=f(a*x+b*y+c) with a,b,c=const. For your example y'=(x+y)^2 just enter: deLinSubst((x+y)^2,x,y) (enter only the RHS, not y'=...) In this simple form you get a solution only if Xcas can 'solve' the equation for y, but it could easily be modified to return at least an implicit solution if the equation is unsolvable. Franz RE: deTaylor - salvomic - 05-22-2015 10:13 PM
(05-22-2015 09:44 PM)fhub Wrote: Well, in the meantime you could also use the following function: well done, Franz! it works: gives the true (general) solution of ODE y'=(x+y)^2 -> TAN(G_0+x)-x Salvo |