Word Count : 450

---

हमें निम्नलिखित अवकल समीकरण का हल फोबेनियस विधि से निकालना है: [ \frac{d^2y}{dx^2} + \frac{1}{2x}\frac{dy}{dx} + y = 0 ] यहाँ (x=0) पर एक नियमित संधि (regular singular point) है, इसलिए हम फोबेनियस विधि का उपयोग करेंगे। फोबेनियस विधि के अनुसार हम हल मानते हैं: [ y = x^r \sum_{n=0}^{\infty} a_n x^n, \quad a_0 \neq 0 ] पहले अवकलन निकालते हैं: [ \frac{dy}{dx} = \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1}, \quad \frac{d^2y}{dx^2} = \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2} ] इन्हें समीकरण में प्रतिस्थापित करें: [ \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2} + \frac{1}{2x} \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1} + \sum_{n=0}^{\infty} a_n x^{n+r} = 0 ] दूसरे टर्म को सरल करें: (\frac{1}{2x} \sum a_n (n+r) x^{n+r-1} = \sum a_n ___ ________ __________ _________ _____.
________ ______ _____ __________ ___ _____ _____ ________ ____.
___ _________ ____ _______ _________ _________ __________.
____ ___ _________ ________ ________ ______.
__________ ______ ____ ___ ________ ___ ___ __________ ______ _______ ________.
____ ______ ___ _________ ________ ___.
___ ____ __________ _________ ______ ____ ___.
__________ _____ __________ _____ _________ _____ __________ ______ __________ __________ ____ _____.
____ __________ ________ ________ ____ ____ __________ __________ __________ _________ ____.
_________ ________ __________ _______ ___ ___ ______.
___ _________ ________ ______ _______ ________ ______ _______.
______ _____ _______ _______ ___ ______ ___ __________ _____ ________ ______.
___ ___ __________ _____ ______ ______ _____ ___ ___ _______.
_____ ______ __________ __________ ___ ____ _____ __________ _______ ______ ___ __________.
_________ ______ _______ _________ ____ ____.
___ ________ ___ _______ ____ ______ _____ ______ ________.
_______ _________ ______ _____ _______ ________ _____ ______ __________.
___ __________ ____ ______ _________ ________ __________ ______ ______.
_______ ______ ______ ______ _________ ____ __________ ______.
________ _____ ____ _________ _____ ___ ______ ___ _______ _________ __________.
___ _________ ____ __________ __________ ______ ______ ___.
___ _______ __________ _________ _________ _____ _______ ___ ____ ___ ______ ________.
_________ _________ ___ ___ ________ _______ __________.
___ _______ ______ ________ __________ _______ _____ __________ _______ _______ ______.
______ _________ ___ __________ ______ ____ _________ _________ _______ _________.
__________ ________ ________ ______ _______.
__________ __________ ______ _________ ________ _____ _____ __________ _____ ______.
__________ ______ ____ _____ _____ ______ _______ ___.
_____ ___ ______ _________ ____ _____ ___ _____ ________.
____ ____ _________ ____ __________ ____ ___ ___ ________ _______.
_________ ____ ___ ____ ___ ____ _____.
___ ________ __________ _________ ______.
_____ __________ ______ ______ ________ ____ _____ __________.
_________ ________ _________ ________ ________ ________ _______ ________ __________ __________.
____ ____ _________ ________ ___ ______ __________ _______ ________ _____.
_________ __________ _____ _____ ________ ______ _______ ________.
________ ______ ____ ______ ______ ___ _________ ____ ___ _______ _____ ____.
______ _______ ___ ___ ________ ________.
___ ___ _________ ______ ___ _________ ____ ____.
Get Full Answer on WhatsApp