How to solve differential equation of series general solution

How do we solve differential equation of series general solution?

For example, I tried to solve this differential equation into series solution, but as you can see in the result, it is not showing like y = C1 x^(1/4)(1 - (1/14)x + (1/616)x^2 + ...) + C2 x^(-1/2) (1 - (1/2)x + (1/40)x^2 + ...) and this is the series general solution that I want to make for the result.

How can make the result in the photo to the form like how I typed right before? Could you please type the command for the solution clearly? Thank you :)

[John D'Errico]

Please do not paste in a picture of your code. Then we need to type in the code by hand. Had you simply pasted in text, then we could more easily help you. Is there a good reason why you want to make it more difficult to get help?

Answer 1

[John D'Errico]

Assuming I typed in that line correctly...

syms y(x)
S = dsolve(8*x^2*diff(y,2)+10*x*diff(y)+(x-1)*y==0,'ExpansionPoint',0, 'Order',8)

S = 

There were TWO independent solutions found. You have TWO unknown conditions, which are typically supplied as boundary or initial conditions. But consider what this does?

syms C1 C2
C1*S(1) + C2*S(2)

ans = 

[J]

I appreciate it! It is working! Also, do you know how to solve differential equation with Method of Frobenius which contains indicial equation, indicial roots, and recurrence relation?

[John D'Errico]

Sorry. That is verging too much on homework, and I did my last homework assignment sometime in the 1980's in grad school. If you make an effort and show what you did, you can ask a specific question on Answers, as you did here. But we are not here to do your homework for you.

https://en.wikipedia.org/wiki/Frobenius_method

[J]

Yes, I was just curious if MATLAB had a default function related to method of Frobenius when solving differential equations!

[John D'Errico]

I believe there is no specific tool to do exactly that.

[J]

Aha I see Thank you!

Is is correct that 'ExpansionPoint',0 represents that the center is 0 which means x naught = 0 and it is correct that 'Order', 8 means the solution terms are 8 terms?