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Topic: Bessel And Airy Functions in Solutions
Replies: 1   Last Post: Apr 4, 2014 9:59 AM

 Axel Vogt Posts: 1,068 Registered: 5/5/07
Re: Bessel And Airy Functions in Solutions
Posted: Apr 4, 2014 9:59 AM

On 04.04.2014 11:53, Axel Vogt wrote:
> On 04.04.2014 06:17, Thomas D. Dean wrote:
>> I have a problem,
>>
>> ode:=diff(y(x),x,x)+(a*x+b)*y(x)=0;
>> raw_soln:=dsolve(ode);
>> soln1:=convert(raw_soln,BesselI);
>>
>> This gives an expression with several terms.
>>
>> Maxima gives a solution,
>>
>> [y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*%k2*sqrt(a*x+b)
>> +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*%k1*sqrt(a*x+b)]
>> where bessel_y is the second kind and bessel_j is the first kind.
>>
>> If I assume %k2 is _C2 and %k1 is _C1
>>
>> maxima := y(x) = BesselY(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*_C2*sqrt(a*x+b)
>> +BesselJ(1/3,2*(a*x+b)^(3/2)/(3*abs(a)))*_C1*sqrt(a*x+b);
>> soln2:=convert(maxima,BesselI);
>>
>> I am having problems determining if these are the same.
>>
>> Ideas?
>>
>> Tom Dean

>
> Better use different constants for Maxima's solution. And convert
> to Airy functions. Then comparing the inputs for the Airy functions
> show different expressions. May be you need 0 < a to get the 'same'
> as your raw_solution
>
> convert(maxima, Airy):
> collect(%, [AiryAi, AiryBi]):
> simplify(%, size);
> maxima2:=%;

You want to compare

eq1:=( ( (a*x+b)/a )^(3/2)/a )^(2/3);
eq1:= simplify(eq);
eq2:=(a*x+b)/a^(2/3);

But these are not the same in general

[eq1, eq2]; subs(b=0, a=-1, x=-1, %); evalc(%);

1/2 1/2
[1/2 + 1/2 I 3 , - 1/2 - 1/2 I 3 ]