Differential Equations | Series solution for a second order linear differential equation.

We find a series solution for a second order linear differential equation.

[alexmonroy]

at 5:58 wasn't it suppose to be 2 and not 6 Im kinda of confused @michael penn?

[munguia]

I learned that after distributing x^2 with the summation, the power index stays at 2 and not at 0, or am I incorrect?

[canaryinc]

i hope my brain will accept that Odd numbers can be 2n + 1 and also 2n - 1 sometimes LOL.
This was very helpful btw, even though getting that final result might be a bit tricky when faced with a problem in real time

[holyshit]

I would reduce it to Riccati
In Riccati equation it is easier to guess particular solution
It is possible that integrals need power series
(1+2x^2)y''+6xy'+2y=0
: y
y''/y = -6x/(1+2x^2) y'/y -2/(1+2x^2) | - (y'/y)^2
y''/y - (y')^2/y^2 = -(y'/y)^2 -6x/(1+2x^2) y'/y -2/(1+2x^2)
(y''y-(y')(y'))/y^2 = -(y'/y)^2 -6x/(1+2x^2) y'/y -2/(1+2x^2)
(y'/y)' = -(y'/y)^2 -6x/(1+2x^2) y'/y -2/(1+2x^2)
Let y'/y = z
z' = -z^2 -6x/(1+2x^2)z-2/(1+2x^2)
Now particular solution is in the form
z_{p} = (ax+b)/(1+2x^2)

[MGoebel-ce]

Everything after 16:30 seems wrong, calculatory error when calculating a5 (should be a 4 instead of a 5 in the denominator). Or i have tomatoes on my eyes. Anyhow very nasty problem in the end, simple but very confusing

[holyshit]

I can record video for non-series solution of this equation but my english is not very good
I have also low quality microphone

[jarikosonen]

It looks possible the result will not convert to elementary functions...
There is something unclear.

[gyanprakashraj]

EASIEST INTEGRATION SUBSTITUTION...SHITT