Foundations — Direction fields and Euler's method — visual - numerical intuition first

4.6.2 · D1 · Maths › Ordinary Differential Equations › Direction fields and Euler's method — visual - numerical int

Is page par kuch bhi assume nahi kiya gaya hai. "Arrows follow karne" se pehle, tumhe exactly pata hona chahiye ki $\frac{dy}{dx}=f(x,y)$ mein har symbol ka picture mein kya matlab hai. Hum inhe ek-ek karke build karte hain, har ek sirf pehle waale ka use karke.

0. Plane aur ek point $(x,y)$

Sab kuch ek flat sheet par hota hai jisme do number lines right angles par cross karti hain.

• Horizontal line ==$x$-axis== hai — left (negative) se right (positive) padho.
• Vertical line ==$y$-axis== hai — neeche (negative) se upar (positive) padho.
• Ek point $(x,y)$ sheet par ek dot hai: $x$ steps right jao, phir $y$ steps upar.

Yeh topic ko kyun chahiye: poora subject is plane mein rehta hai. Ek "solution" is par draw ki gayi ek curve hogi, aur ODE iske har single dot ke baare mein kuch kehti rahegi.

1. Ek curve, aur letter $y$ do kaam ek saath karta hai

Ek subtlety dhyan se dekho jo sabko trip karati hai. $\frac{dy}{dx}=f(x,y)$ mein letter $y$ ek saath do cheezein mean karta hai, aur tumhe dono pakad ke rakhni hain:

• $y$ ek coordinate ke roop mein — bas ek dot ki height, jaise §0 mein.
• $y$ ek function $y(x)$ ke roop mein — ek rule jo har $x$ ke liye ek height deta hai. Iska graph ek curve hai.

Yeh topic ko kyun chahiye: ek "solution curve" kisi unknown function $y(x)$ ka graph hai. ODE us function ke baare mein ek clue hai; direction fields humein clue dekhne deti hain jab tak hum function nahi jaante.

2. Slope — poore show ka star

Ek straight line par do points chuno. Left wale se right wale tak chalo.

• Run kitna right chale: $\Delta x$ (symbol $\Delta$, "delta," ka matlab sirf change in hai).
• Rise kitna upar chade: $\Delta y$ (negative agar neeche gaye).

Yeh topic ko kyun chahiye: poori ODE slope ke baare mein ek statement hai. $f(x,y)$ ek slope hoga. Euler's method ek slope ko run se multiply karega aur rise paayega. Yeh pakad lo aur baaki arithmetic hai.

3. Average slope se derivative $\frac{dy}{dx}$ tak

Slope §2 ko do points chahiye the. Lekin ek curve bend karti hai — uski steepness jagah-jagah badlti rehti hai. Yeh single point par kitni steep hai?

Yeh topic ko kyun chahiye: $\frac{dy}{dx}=f(x,y)$ ka left side tangent ka slope hi hai. Toh equation literally kehti hai: "$(x,y)$ par meri solution ki tangent $f(x,y)$ amount se tilt karti hai." Yahi sentence direction field hai.

4. Do inputs wala function: $f(x,y)$

§1 mein ek function ek number khata tha. Ab ek aisa function dekho jo do khata hai.

Yeh topic ko kyun chahiye: ODE ka right side yahi machine hai. Koi bhi dot daalo, yeh wahan tilt batata hai draw karne ke liye. Ek grid of dots par aisa karo aur tumne direction field paint kar diya.

5. Sab mila ke — $\frac{dy}{dx}=f(x,y)$ ek picture ke roop mein kya kehta hai

Ab har symbol earn ho gaya hai. Equation dhyaan se padho:

"First-order" kyun: sirf pehla derivative $\frac{dy}{dx}$ appear karta hai — koi $\frac{d^{2}y}{d{x}^2}$ ya usse upar nahi. Woh "1" isliye hai ki ek initial dot $(x_0,y_0)$ ek unique curve pin karne ke liye kaafi hai (Existence and Uniqueness (Picard–Lindelöf) mein precisely bataya gaya hai).

6. Step size $h$ aur index $n$

Field ko numerically walk karne ke liye humein do aur shorthand chahiye.

• ==Step size $h$== : ek chhota fixed run — har step mein hum kitna right move karte hain. Yeh §2 ka $\Delta x$ hai, ek baar choose karo aur rakho.
• ==Index $n$== : ek counting label. $x_0,x_1,x_2,\dots$ stops hain; $y_0,y_1,\dots$ unki heights. Subscript bas kaunsa stop batata hai. Toh $x_{n+1}=x_n+h$ ka matlab hai "agla stop current se ek step $h$ right hai."

Yeh topic ko kyun chahiye: Euler's rule $y_{n+1}=y_n+hf(x_n,y_n)$ hai "new height = old height + run $h$ × slope $f$." Isme ab har symbol ka ek picture hai: §2 run × slope = rise deta hai, §3–4 slope deta hai, §6 ladder deta hai.

7. Bending term $y^{\prime \prime }$ (kyun Euler sirf approximate hai)

Parent mein error explain karte waqt ek aakhri symbol aata hai.

Prerequisite map

Equipment checklist

Right side cover karo aur khud test karo.

Connections

• Taylor's Theorem — "tangent ke saath step" ko exact statement mein convert karta hai $\frac{h^2}{2}{y}^{\prime \prime }$ remainder ke saath.
• Existence and Uniqueness (Picard–Lindelöf) — kyun ek dot $(x_0,y_0)$ ek solution curve fix karta hai.
• Separable First-Order ODEs — pehli family jiske exact curves ko tum field ke against compare kar sakte ho.
• Runge-Kutta Methods (RK4) — Euler ke undershoot ko beat karne ke liye har step mein kayi slopes sample karo.
• Stability and Stiff Equations — jahan sirf left slope padhna walk ko explode kar deta hai.