Non-homogeneous — variation of parameters: Question bank

4.6.15 · D5 · Maths › Ordinary Differential Equations › Non-homogeneous — variation of parameters

Yahan jo vocabulary reuse hoti hai, wo kabhi surprise nahi honi chahiye:

$y_{1}, y_{2}$ — homogeneous equation $y^{''} + p y^{'} + q y = 0$ ke do independent solutions.
$g (x)$ — standard-form equation ka right-hand side (leading coefficient $1$ ).
$W = y_{1} y_{2}^{'} - y_{2} y_{1}^{'}$ — Wronskian, "kya ye dono genuinely alag hain?" ka scale.
$y_{p} = u_{1} y_{1} + u_{2} y_{2}$ — constants ko vary karne se bana particular solution.

Why questions

Hum exactly ek extra condition kyun impose karte hain, zero ya do kyun nahi?

Hamare paas do unknown functions hain lekin sirf ek equation (the ODE) hai, toh ek degree of freedom genuinely free hai — exactly ek condition use hoti hai use bina system ko over-constrain kiye.

Tamam possible conditions mein se specifically

u_{1}^{'} y_{1} + u_{2}^{'} y_{2} = 0

kyun choose karte hain?

Ye precisely un terms ko cancel karta hai jinki agla derivative

u_{1}^{''}, u_{2}^{''}

produce karta, surviving system ko

u_{1}^{'}, u_{2}^{'}

mein first-order aur linear rakhta hai — algebraically sabse simple possible outcome.

Equation ko

g

padhne se pehle standard form mein kyun laana zaroori hai?

Derivation ne terms ko

y^{''} + p y^{'} + q y = g

ki tarah group kiya tha; vanishing brackets

y_{i}^{''} + p y_{i}^{'} + q y_{i}

tabhi zero hote hain jab

p, q

normalized coefficients hon, jo

g

ko leading coefficient se divide karne ke baad RHS hone par force karta hai.

Wronskian denominator mein kyun aata hai,

y_{1} + y_{2}

jaise sum mein kyun nahi?

u_{1}^{'}, u_{2}^{'}

ke linear system mein coefficient matrix ka determinant hai, toh Cramer's Rule se solve karne par naturally us determinant se divide hota hai, jo exactly

W

hai.

VoP ko "variation of parameters" kyun kehte hain?

Parameters wo constants

c_{1}, c_{2}

hain jo general homogeneous solution mein hote hain; hum unhe vary karte hain —

x

ke functions banaate hain — isliye "variation of parameters".

VoP principle mein wahan kyun hamesha succeed karta hai jahan undetermined coefficients fail ho sakta hai?

VoP

y_{p}

ko known quantities ke integration se produce karta hai, aur ek continuous function ka antiderivative hamesha exist karta hai; undetermined coefficients ek form guess karta hai, jo sirf special catalogue ke

g

ke liye exist karta hai.

Non-zero Wronskian guarantee kyun karta hai ki method kaam karega?

W \neq = 0

certify karta hai ki

y_{1}, y_{2}

linearly independent hain, toh ye genuine basis form karte hain aur

u_{1}^{'}, u_{2}^{'}

ke liye

2 \times 2

system ka har point par unique solution hota hai.

Edge cases

Agar

g (x) \equiv 0

ho toh kya hoga?

Tab

u_{1}^{'} = u_{2}^{'} = 0

, toh

u_{1}, u_{2}

constants hain aur

y_{p}

wapas complementary function mein collapse ho jaata hai — VoP sahi tarah report karta hai ki homogeneous equation ke liye "koi extra particular part needed nahi hai".

Agar

y_{1}

aur

y_{2}

accidentally same solution hon (ya proportional hon) toh kya hoga?

Tab

W \equiv 0

har jagah hoga, formula zero se divide karega, aur method break ho jaayega — red flag ki tumhare paas actually Second-order linear homogeneous ODE ke do independent solutions nahi hain.

Agar repeated-root case mein

y_{1} = e^{r x}, y_{2} = x e^{r x}

mile toh kya hoga?

Bilkul theek hai — ye independent hain (

W = e^{2 r x} \neq = 0

), toh VoP unchanged apply hota hai; repeated root sirf is baat ko affect karta tha ki tumne

y_{1}, y_{2}

kaise dhundhe, method ko nahi.

Agar kisi point par

g (x)

mein jump ya singularity ho (jaise

g = 1/ x

x = 0

) toh kya hoga?

u_{1}, u_{2}

ke integrals diverge ho sakte hain ya solution sirf us point se bachne wale interval par valid ho sakta hai; VoP sirf wahan guaranteed hai jahan

p, q, g

continuous hon aur

W \neq = 0

ho.

Agar

y_{p}

mein

y_{1}

y_{2}

ke proportional koi term aaye, toh kya ye galti hai?

Nahi — aisa term ek legitimate homogeneous piece hai aur

c_{1} y_{1} + c_{2} y_{2}

mein absorb ho sakta hai; general solution affected nahi hota, jaise Example 2 mein

- e^{x}

c_{1} e^{x}

mein merge kiya.

Kya VoP higher-order ya non-constant-coefficient equations tak extend ho sakta hai?

Haan — order

n

ke liye

n - 1

zero-conditions impose karte hain aur higher-order Wronskian ke saath

n \times n

Cramer system solve karte hain; non-constant coefficients theek hain jab tak standard form mein divide na kar lo. Ye viewpoint Green's function ko bhi underlie karta hai.

Recall Traps ka one-line summary

$g$ se pehle standard form; minus sirf $u_{1}^{'}$ par; $W$ denominator mein aur kabhi zero nahi; $+ C$ drop karo; aur yaad raho Condition 1 ek choice hai, koi law nahi.

Connections

Wronskian — fairness scale; $W \neq = 0$ proceed karne ka licence hai.
Cramer's Rule — kyun $W$ un signs ke saath denominator mein aata hai.
Method of Undetermined Coefficients — guess-based rival jo exotic $g$ par VoP se peeche rehta hai.
Second-order linear homogeneous ODE — $y_{1}, y_{2}$ supply karta hai.
Reduction of Order — jab sirf ek known ho toh doosra $y_{2}$ kaise paayein.
Green's function — is poore idea ka integral-kernel generalization.

Question bank — Non-homogeneous — variation of parameters

True ya false — justify karo

Error dhundho

Why questions

Edge cases

Connections