4zy''+2y'+y=0
find series solution about z=0 and identify these solution in closed form
If you divide through by x you get . The coefficient function for is Laurent series of order and same for . This means the equation is solvable through Frobenius method. The Frobenius equation would be so . These do not differ by integers so you can solve this equation on by letting where . For each value you will get a new linearly independent solution.
Hello,
Let
You will have to determine ~
Putting all the powers to , by changing the indices ~
In the first and second ones, n-1 becomes n. Then the series will start at 1 and 0..
Now, starting the series at the same point :
Therefore :
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And :
Find the induction formula