deTaylor

05222015, 01:24 PM
(This post was last modified: 05222015 01:39 PM by fhub.)
Post: #1




deTaylor
Here's a small CAS function deTaylor for the HPPrime (similiar to deSolve) which I've originally written for Xcas.
It approximately solves 1st and 2ndorder differential equations (with initial conditions) as nth 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 Primeemulator commandline in CASmode) Code:
Example 1: y'=x*y^2+1 with y(0)=1 (6thdegree 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 

05222015, 06:10 PM
(This post was last modified: 05222015 06:10 PM by salvomic.)
Post: #2




RE: deTaylor
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(xc)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 ∫aL√0mic (IT9CLU), HP Prime 50g 41CX 71b 42s 12C 15C  DM42 WP34s :: Prime Soft. Lib 

05222015, 08:27 PM
Post: #3




RE: deTaylor
(05222015 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 

05222015, 08:31 PM
Post: #4




RE: deTaylor
(05222015 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 ∫aL√0mic (IT9CLU), HP Prime 50g 41CX 71b 42s 12C 15C  DM42 WP34s :: Prime Soft. Lib 

05222015, 09:44 PM
(This post was last modified: 05222015 09:48 PM by fhub.)
Post: #5




RE: deTaylor
(05222015 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 

05222015, 10:13 PM
Post: #6




RE: deTaylor
(05222015 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 ∫aL√0mic (IT9CLU), HP Prime 50g 41CX 71b 42s 12C 15C  DM42 WP34s :: Prime Soft. Lib 

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