On Saturday, 9 January 2021 at 12:35:14 UTC+11,
drhu...@gmail.com wrote:
On Friday, 8 January 2021 at 10:36:30 UTC+11, drhu...@gmail.com wrote:
On Thursday, 7 January 2021 at 23:45:38 UTC+11, drhu...@gmail.com wrote:
On Thursday, 7 January 2021 at 23:28:50 UTC+11, drhu...@gmail.com wrote:
On Thursday, 7 January 2021 at 13:30:51 UTC+11, Nasser M. Abbasi wrote:
why wolfram cannot find simple solution in many equations? many examples, e.g.:
1/x*y"-y-x^2=0
1/x*y"-y-x^3=0
1/x*y"-y-x^4=0
......
mathHand.com can: http://server.drhuang.com/input/?guess=dsolve%281%2Fx*ds%28y%2Cx%2C2%29-y-x%5E3%29
so it is good idea to use two software to check solution.
For the first ode above. This is inhomogeneous Airy ode. The solution you give for
1/x*y''[x]-y[x]-x^2==0
is
y(x)= -6+Ai(x)*C1+Bi(x)*C2-x^3
But this solution is not verified by Maple:
==============
restart;
ode := 1/x*diff(y(x),x$2) - y(x) - x^2 = 0;
sol := y(x) = -6+AiryAi(x)*_C2 + AiryBi(x)*_C1-x^3;
odetest(sol,ode);
x^3 - x^2
=================
Which is not zero. Hence not a valid solution?
--Nasser
the bug fixed. thanks.
mathHand.com
input your equation into mathHand.com, click the "dsolve" button to solve, then click the "test" button to test its solution, or click the "to inte" button to plot if it did not auto plot.
http://mathHand.com
wolfram gives very complicated solution:
y"-x y-x^3+2=0
mathHand.com gives very simple solution: http://server.drhuang.com/input/?guess=dsolve%28ds%28y%2Cx%2C2%29-x*y-x%5E3%2B2%29
wolfram gives very complicated solution:
y"-x y'-x y-x^3+2=0
mathHand.com gives very simple solution: http://server.drhuang.com/input/?guess=dsolve%28ds%28y%2Cx%2C2%29-ds%28y%2Cx%2C1%29*x-y*x-x%5E3%2B2%29&inp=ds%28y%2Cx%2C2%29-ds%28y%2Cx%2C1%29*x-y*x-x%5E3%2B2
Wolfram cannot find simple solution but mathHand.com can in following equations:
y"-x*y'-x*y-x=0
y"-x*y'-x*y-x^2=0
y"-x*y'-x*y-x^3+2=0
y"-x*y'-x*y-x^4-6=0
y"-x*y'-x*y-x^5+12=0
y"-x*y'-x*y-x^n=0, where n=1,2,3 ......
y"-a*x*y'-b*x*y-c*x^n=0, where n=1,2,3 ......
y"-y'-x*y-x=0
y"-y'-x*y-x^2=0
y"-y'-x*y-x^2-1=0
y"-2*y'-x*y-x^2-2=0
y"-4*y'-x*y-x^2-4=0
y"-y'-x*y-x^n=0, where n=1,2,3......
y"-a*y'-b*x*y-c*x^n=0, where n=1,2,3 ......
y"-x*y-x^3+2=0
y"-x*y-x^6+40=0
y"-x*y-x^n=0, where n=1,2,3 ......
......
more examples in
http://drhuang.com/science/mathematics/fractional_calculus/fractional_differential_equation.htm
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