homogeneous DE .. x^2 y" + xy' + (x^2 - 1/4)y = 0...

Question

anonymous · Answer

assume that the solution in the form of  $$y = \sum_{0}^{\infty} a _{n} x ^{n+r} $$

anonymous · Answer

r is constant value whose value is yet to be determined..

anonymous · Answer

show that the two equation to be solved are as follows : $$a _{0} ( c ^{2} - 1/4 ) = 0 ,     a _{1} [ (1+c)^{2} - 1/4 ] = 0 ...$$

anonymous · Answer

@amistre64  , @SithsAndGiggles  i need your help..

anonymous · Answer

x3 y(x) y' (x2 - 1/4) y'' (x) = 0

dumbcow · Answer

are you sure the solution is a polynomial? http://www.wolframalpha.com/input/?i=x%5E2+y%22+%2B+xy%27+%2B+%28x%5E2+-+1%2F4%29y+%3D+0

dumbcow · Answer

oh sorry nevermind i didn't see it was an infinite series

anonymous · Answer

:).. so how to solve this ?

kenljw · Answer

Its not an infinite series, if y is the form EXP(ax) the y' = aEXP(ax), and y'' = a^2EXP(ax). Place these in equation and you end up with a quadratic equation.  Solve for x and you end up with 2 Eigen values to be place in 
EXP() for solution.

kenljw · Answer

Noteworthy Eigen is German for characteristic, that why there called characteristic equation.  Germans where great mathematicians and the use of Eigen was to their Honor,