Math and Differential Equations Exercise I need help with just number 4. It requires series solutions ordinary points. please show the steps along with the

Math and Differential Equations Exercise I need help with just number 4. It requires series solutions ordinary points. please show the steps along with the final answer. I started it and will attach how i started it to show, but i do not know how to continue from there –
o=h&+, 11-X) h
n-o
n=1
n=2
=> XY” – Y+3y0
y = Z cnxn , y’=Ž cn nxnol, y”= 8 cnn (n-1) xn – 2
2 38
(x-1) E Cnn(n-1) xn 2 + 32 cnxn
XE Cnncn-1) xn--I Cnncn- xn- 2 + 3 E Cnxn
nao
= { Con(n-1) xn-1 § cnncn-1) x 2 + 3 cnxn
n=2
n.2
–
no
n.2
n = 2
1. 2x’y'”‘ +19x?y” +39xy’ +9y = 0 Hint: You will need to first determine the auxiliary
equation for a third-order equation.
2. x’y” – 4xy’ +6y= 2×4 + x2
3. 2xy” + y’ + y = 0
4. (x-1)y” + 3y = 0

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