4.6.17 · D5 · HinglishOrdinary Differential Equations
Question bank — Power series solutions — ordinary points
4.6.17 · D5· Maths › Ordinary Differential Equations › Power series solutions — ordinary points
Shuru karne se pehle, teen words jo hum baar baar use karte hain — yahan pin kar diye hain taaki koi symbol unexplained na rahe:
True or false — justify
A rational function is analytic at exactly when
True — polynomials ka ratio har jagah convergent Taylor series rakhta hai jahan denominator nonzero ho; ka sabse paas ka zero hi sabse paas ki singularity hai.
Agar leading coefficient in at nonzero hai, toh ek ordinary point hai
True (polynomial coefficients ke liye) — se divide karne par aur wahan finite aur analytic rehte hain, isliye dono analytic hain.
Har ordinary point exactly do linearly independent power-series solutions deta hai
True — yeh Fuchs' theorem ka ordinary point par promise hai; aur do independent branches ko seed karte hain (dekho Existence and uniqueness for linear ODEs).
Series solution ki radius of convergence nearest singular point ki distance se choti ho sakti hai
False — Fuchs guarantee karta hai ki yeh kam se kam utni distance hai; yeh zyada ho sakti hai (agar singularity removable ho) lekin kabhi guaranteed choti nahi hogi.
Ordinary point ke baare mein power series solution hamesha sab real ke liye converge karta hai
False — sirf tab jab koi finite singular point na ho (jaise Hermite/Airy). ke liye ke baare mein odd series sirf ke liye converge karta hai.
Agar ek series ek interval par identically zero hai, toh uske kuch coefficients phir bhi nonzero ho sakte hain
False — ek convergent power series jo ek interval par zero hai uska har coefficient zero hota hai; yahi identity "har coefficient ko set karo" ko allow karti hai.
Recurrence relation aur ko determine karta hai jab ODE pata ho
False — recurrence ko earlier coefficients se link karta hai, isliye yeh ko kabhi touch nahi karta; woh do free integration constants hi rehte hain.
Do series ko term-by-term add kar sakte ho even agar unke powers of offset hain
False — coefficients sirf power-by-power combine ho sakte hain; pehle re-index karo taaki har sum ban jaye (dekho Recurrence relations).
Spot the error
" is singular for kyunki appear karta hai."
Error — classification , use karta hai; par denominator hai, isliye ordinary hai. Singularities par hain.
"Main seedha mein substitute karta hoon, toh mujhe kabhi standard form ki zaroorat nahi."
Half-error — tum undivided form mein substitute karte ho (theek hai), lekin point ko classify karne ke liye standard-form pehle dekhna padega, warna tum nahi jaanoge ki point ordinary hai ya convergence kahan rukti hai.
" mein re-index karne se milta hai."
Error — lower limit bhi shift hoti hai: , isliye sum se start hota hai. rakhne se aur terms chupke se drop ho jaate hain aur corrupt ho jaate hain.
"."
Error — aur terms factor carry karte hain, isliye kuch contribute nahi karte; sum honestly se start hota hai. likhna value mein galat nahi hai lekin index-mismatch mistakes ko invite karta hai.
"Recurrence dono series terminate karta hai kyunki zero deta hai."
Error — sirf even branch ko zero karta hai ( ke through). Odd branch odd par chalta rehta hai, aur wahan kabhi zero nahi hota, isliye yeh ek infinite series hai.
"Substitute karne ke baad mujhe mila, toh main poore bracket-sum ko zero set karta hoon."
Error — tum har coefficient bracket ko individually zero set karte ho, ek equation per . Unhe ek equation mein sum karne se recurrence ke liye zaroori saari information kho jaati hai.
"Kyunki free hain, main bhi freely choose kar sakta hoon."
Error — jab ek baar choose ho jaata hai, recurrence se forced ho jaata hai ( for ). Sirf pehle do free hain; ek 2nd-order ODE mein exactly do constants hote hain.
Why questions
Hum re-index kyun karte hain instead of sums ko as-is add karne ke?
Kyunki same label ke under powers contribute karte hain; re-indexing dummy ko rename karta hai taaki identical powers line up ho sakein aur legally coefficient-by-coefficient add ho sakein.
aur ki analyticity check kyun karte hain (na ki ki)?
Theorem ki hypothesis equation ke coefficients par hai; agar analytic hain, toh conclusion yeh hai ki ek well-behaved analytic solution exist karta hai. ko advance mein check nahi kar sakte — wohi toh hum solve kar rahe hain.
Power series substitute karne se differential equation algebra mein kyun convert ho jaata hai?
ko differentiate karna sirf indices shift aur scale karta hai, isliye ODE coefficients ke beech ek relation ban jaata hai — recurrence — jisme koi derivatives nahi bachte (yahi is poore method ka existence ka reason hai).
Legendre-type equations kabhi kabhi polynomial (terminating) solutions kyun dete hain?
Recurrence numerator mein jaisa ek factor hota hai; jab hota hai toh agla coefficient zero ho jaata hai aur chain rukk jaati hai, ek finite polynomial bacha jaata hai — the Legendre polynomials.
Singular point par hum yeh ordinary-point recipe kyun use nahi kar sakte?
Singular point par ya analytic nahi hota, isliye plain series converge nahi kar sakti ya koi solution miss kar sakti hai; tumhe Singular points and Frobenius method ke fractional/log powers chahiye.
Hermite aur Airy equations hamesha sab ke liye convergent solutions kyun dete hain?
Unke standard-form polynomials hain, isliye poori real line par analytic hain bina kisi finite singular point ke, toh guaranteed radius of convergence infinite hai (dekho Hermite and Airy equations).
Substitution ke dauraan polynomial coefficients jaise ko sums ke andar kyun rakhte hain?
se series multiply karna uske powers shift karta hai; pehle coefficient distribute karo, phir har resulting sum ko re-index karo, taaki har term ki power explicit rahe aur koi term drop na ho.
Edge cases
Kya automatically har ODE ke liye ordinary hai?
Nahi — sirf tab jab wahan analytic hon. ke liye blow up karta hai, isliye singular hai despite innocent dikhne ke.
Kya agar analytic ho lekin nahi?
Tab bhi singular point hai — definition require karti hai ki dono aur analytic hon; ek failure point ko disqualify karne ke liye kaafi hai.
Kya hota hai agar ek branch terminate kare aur doosra nahi?
Tumhe ek polynomial solution milti hai aur ek genuine infinite series; saath mein yeh phir bhi do linearly independent solutions hain, exactly jaisa Fuchs promise karta hai.
Agar nearest singular point ek complex number ho, toh kya yeh real series ko affect karta hai?
Haan — radius of convergence complex plane mein nearest singularity ki distance hai, isliye par ek complex singular point convergence ko tak cap kar deta hai even real line par.
Kya agar ho? Kya method change hota hai?
Nahi — substitute karo taaki ordinary point par aa jaye, expand karo, same recipe chalao, phir translate back karo.
Kya ek first-order ODE is tarah solve ho sakta hai, aur kitne constants appear hote hain?
Haan, ek single free constant ke saath — recurrence ko se fix karta hai, sirf free rehta hai, jo first-order equation ke ek constant se match karta hai.
Kya agar dono aur hon?
Tab recurrence har force karta hai, trivial solution deta hai — consistent hai, kyunki linear homogeneous ODE ke liye ke saath ek hi solution zero hoti hai.
Recall One-line summary of the traps
== in standard form se classify karo, re-index karo lower limit including, har coefficient ko zero set karo, aur yaad rakho sirf free hain. Convergence nearest (possibly complex) singularity== tak pahunchti hai.