Classification — order, degree, linear vs nonlinear, autonomous vs non-autonomous: Question bank

4.6.1 · D5 · Maths › Ordinary Differential Equations › Classification — order, degree, linear vs nonlinear, autonom

Shuru karne se pehle, teen vocabulary anchors taaki yahan koi word bina explanation ke na aaye:

Definition Woh words jo hum test karte hain

Order = kaun sa derivative sabse gehra hai (ek $y^{''}$ ek $y^{'}$ se bada hota hai). Socho "kitne nested rates hain."
Degree = us sabse gehre derivative par power kya hai, lekin tabhi jab tumne equation ko sabhi derivatives mein polynomial form mein saaf kar liya ho. "Saaf karna" precise hai: (a) har woh denominator clear karo jisme derivative ho (multiply through karo), aur (b) derivative par se har radical sign hatao (dono sides ko matching power tak raise karo), aur tabhi sabse gehre derivative par exponent padho. Agar aisa koi polynomial form exist nahi karta (jaise koi $sin$ ya $ln$ derivative ka bach jaaye), to degree undefined hai.
Linear = unknown $y$ aur uske derivatives seedhe aate hain: first power mein, kabhi aapas mein multiply nahi, kabhi nonlinear function mein wrap nahi. Aage ke coefficients (independent variable ke functions) chahe kitne bhi bure hon, chalega.
Autonomous = independent variable kabhi khud se right-hand side par nahi aata; rule sirf is baat par depend karta hai ki tum kahan ho, is baat par nahi ki ghadi kya dikha rahi hai.

WHY wale sawaal

Yeh har check ke reason ko probe karte hain, parent ke "WHY" emphasis se match karte hue. Do mein step figure hai taaki intuition dikhti hai, sirf batayi nahi jaati.

Degree padhne se pehle radicals aur denominators kyun clear karne chahiye?

Radical ya fraction sachi power ko chhupaata hai:

1 + (y^{'})^{2}

mein koi honest exponent nahi dikh raha. Squaring (aur koi denominator clear karna) polynomial ko saaf kar deta hai, toh degree sirf cleaned form se padhi jaati hai.

Linearity, autonomy nahi, yeh kyun decide karta hai ki superposition apply hoga ya nahi?

Superposition ko machine

L

ki zaroorat hoti hai jo sums aur scalars par distribute kare —

L [c_{1} y_{1} + c_{2} y_{2}] = c_{1} L [y_{1}] + c_{2} L [y_{2}]

— jo bilkul linearity hai. Autonomy explicit-

t

dependence ke baare mein hai aur is distributivity ko kabhi touch nahi karta. Figure neeche.

Order kyun batata hai ki kitni initial conditions deni hain? ::: Har derivative ko integration se undo karne par ek free constant aata hai; order- $n$ equation mein $n$ constants chhupe hote hain, toh unique solution pin down karne ke liye $n$ conditions chahiye. Autonomous equations ko phase line se kyun study kiya ja sakta hai jabki non-autonomous generally nahi? ::: Autonomous $y^{'} = f (y)$ mein slope sirf $y$ par depend karta hai, toh slope field ke arrows har vertical column mein identical hote hain — tum poori picture ko ek $y$ -axis (phase line) par collapse kar sakte ho aur flow padh sakte ho. Explicit $t$ ke saath arrows column by column badal jaate hain, toh ek static 1-D picture motion capture nahi kar sakti — dekho Phase Line and Equilibria. Figure neeche.
Linearity ko "sabse important split" kyun kaha jaata hai? ::: Linear ODEs superposition obey karte hain aur unka ek complete solution theory hai (integrating factors, characteristic equations); nonlinear wale generally nahi karte, toh yeh single label sabse zyada change karta hai ki kaunse tools legal hain, jaise Separable Equations vs Second-Order Linear ODEs with Constant Coefficients. $x^{2}$ ya $e^{x}$ jaese bure coefficients linearity kyun nahi todte? ::: Linearity ek statement hai unknown $y$ ke baare mein: use first power mein aana chahiye, unmultiplied, unwrapped. Coefficients woh "landscape" hain jisme equation rehti hai aur woh independent variable ke koi bhi function ho sakte hain. Autonomous solutions "time-translates" ek doosre ke kyun hote hain? ::: Agar rule $f (y)$ independent variable ka kabhi zikr nahi karta, toh ek solution $y (t)$ ko $y (t + c)$ par shift karna phir bhi wohi rule satisfy karta hai, kyunki equation nahi bata sakti ki ghadi hila di gayi. Toh kisi solution ka har horizontal shift phir bhi ek solution hai — oopar ki figure mein identical arrow-columns ke roop mein dikh raha hai.

Edge cases

Boundary aur degenerate inputs jo naive checker bhool jaata hai.

"Equation"

y^{'} = 0

ko classify karo.

Order

1

, degree

1

, linear (zero forcing wali linear equation, yaani homogeneous

g = 0

), aur autonomous (koi explicit independent variable nahi). Iske solutions sab constants hain —

y

ki har value ek equilibrium hai.

Kya purely algebraic equation jaise

y^{2} = x

(koi derivatives nahi) ek ODE hai?

Nahi. Koi derivative present nahi hai toh koi "highest derivative" nahi hai, toh order undefined hai aur yeh differential equation nahi hai — yeh ek algebraic relation hai.

(\frac{d ^{2} y}{d x ^{2}})^{1/3} = y

ka order aur degree kya hai?

Cube both sides first to clear the fractional power:

(\frac{d ^{2} y}{d x ^{2}}) = y^{3}

. Ab order

2

, degree

1

. Exponent

1/3

ek disguise tha, derivative ka genuine cube-root nahi.

Kya

y^{''} + ω^{2} y = 0

linear hai jabki

ω^{2}

power jaesa lagta hai?

Haan.

ω

ek constant hai, unknown nahi;

ω^{2}

sirf

y

ko multiply karne wala ek number hai. Sirf

y

ya uske derivatives ki powers linearity ko khataraa deti hain.

y^{'} + y = e^{t}

ko classify karo: linear aur autonomous, ya kuch aur?

Linear aur non-homogeneous (

g (t) = e^{t}

) lekin non-autonomous, kyunki

e^{t}

se independent variable explicitly aata hai. Linear aur autonomous alag-alag axes hain.

Ek "degree undefined" equation — kya uska order phir bhi well-defined ho sakta hai?

Haan.

sin (y^{''}) + y = 0

ki degree undefined hai phir bhi uska sabse gehra derivative clearly

y^{''}

hai, toh order

2

hai. Order bach jaata hai jab degree fail ho jaati hai.

Kya ek equation simultaneously nonlinear, non-autonomous, aur degree greater than one ki ho sakti hai?

Haan — charon labels independent hain.

x (y^{''})^{3} + (y^{'})^{2} = sin x

order

2

, degree

3

, nonlinear (derivatives ki higher powers), aur non-autonomous (explicit

x

) hai.

Recall Jaane se pehle ek-line self-test

Cover the answers and re-derive: $y y^{''} = sin x$ ke liye — order? degree? linear? autonomous? homogeneous? ::: Order $2$ , degree $1$ (the $y^{''}$ appears to first power), nonlinear (product $y \cdot y^{''}$ ), non-autonomous (explicit $x$ in $sin x$ ), aur non-homogeneous ( $g (x) = sin x \neq = 0$ ).

Question bank — Classification — order, degree, linear vs nonlinear, autonomous vs non-autonomous

Sahi ya Galat — justify karo

Galti pakdo

WHY wale sawaal

Edge cases