Power series soln for 4xy"+3y'+3y=0 near origin for x>0

[jerina16]

hi, may somebody help me to answer this.... i really dont know how to start.

1. obtain the power series solution of 4xy" + 3y' +3y = 0 near the origin for x > 0.

2. Find two linearly independent solutions, valid for x >0, of xy" + y = 0

3. Obtain two linearly independent solutions, valid for x > 0, of x2y'' - x(1 + x)y' + y = 0

tnx!

 

[tkhunny]

Re: Power series

1.

a. Write an expression with arbitrary coeficients. y = infinite sum of stuff
b. Find the necessary derivatives. y' = infinite sum of stuff, but differentiated term-by-term AND y'' = well, you get the picture
c. Apply the structure of the given differential equation.
d. Reconcile the indeces.
e. Put it all together in one summation.
f. With some initial conditions, you should be done.
g. If it wasn't FUN, you must have done somethign wrong. Can you even imagine such a powerful technique as this? There IS a solution to your differential equation!
h. Of course, there are convergence issues, but that would spoil our fun so I won't say more of that.