4.8.23 · D1Modified Euler (Heun's method)
Ek curved path pe aage badhne ke liye, sirf abhi ki direction pe bharosa mat karo — dekho ki step ke end pe direction kya hogi, phir un dono directions ka average lekar chalo. Modified Euler (Heun's method) mein jo bhi hai, woh bas yahi "do slopes ka average" wala idea symbols mein likha hua hai.
Is page pe assume kiya gaya hai ki tumne kuch nahi dekha. Yahan tak ki d x d y = f ( x , y ) jab uski simple-words mein meaning aur ek picture ho jaaye.
x y == x == == y == point ( x , y ) x y
Figure 1 dekho. Horizontal chalk line x y path .
f ( x , y ) f machine hai: tum isme apni current position x y slope hai — jahan tum khade ho wahan path kitni steeply upar ya neeche ja rahi hai. f ( 0 , 1 ) = 1 x = 0 1 1
Topic ko f ( x , y ) f ( x ) dono pe depend kar sakta hai — tum kahan ho aur kitne upar ho. Ek jagah par hill ki steepness tumhari altitude ke hisaab se badal sakti hai, isliye machine ko dono inputs chahiye.
Definition Slope — "rise over run"Slope ka jawab: agar main thoda sa right mein step karun, toh main kitna upar jaunga? Agar right mein 1 2 2 1 0.4 neeche jao, toh slope − 0.4
Figure 2 dekho. Tangent line ke neeche chhote right triangle mein ek horizontal leg hai (run ) aur ek vertical leg hai (rise ). Slope = run rise
Intuition "Slope" hi poora game kyun haiNumerical methods ek curve step by step banate hain, aur har step pe unke paas sirf slope ki information hoti hai. Bada positive slope matlab "agle step mein steeply upar charhna hai"; zero slope matlab "abhi flat rehna hai"; negative matlab "neeche utarna hai." Agar tum slope padh sako, toh tum path ka agla chhota piece predict kar sakte ho.
Sign cases jo tumhe yaad hone chahiye:
Slope > 0
Slope = 0
Slope < 0
Parent note ke Example 2 mein, start slope f ( 0 , 1 ) = 0 exactly zero , matlab curve start se flat nikalta hai phir neeche jhukta hai. Yeh "slope = 0
d x d y d d y d x d x d y rise-over-run ko ek single point tak shrink karna hai — path ka slope bilkul wahan jahan tum khade ho.
y ′ == y ′ == d x d y y y ′ = f ( x , y ) "path ka slope woh number hai jo rule f Woh ek line hi woh problem hai jo poora chapter solve karta hai.
Topic ko "instant pe slope" ke liye special symbol kyun chahiye? Kyunki curve ka slope point se point pe badalta hai. Ek bade run pe rise-over-run ek average deta hai; derivative ek exact jagah ka slope deta hai, jo next step aim karne ke liye chahiye.
d x d y Kyun sahi lagta hai: yeh b a Fix: ise ek symbol samjho jiska matlab hai "instantaneous slope." d y d x
d x d y = f ( x , y ) har point pe, meri path ka slope woh number hona chahiye jo machine f Yeh directly height nahi batata — sirf direction jo har spot pe leni hai. Tumhara kaam hai slope rule se har jagah ki height recover karna.
Intuition Ek slope rule kaafi kyun nahi hai — ek starting height chahiyeInfinitely many paths ek hi slope rule rakhte hain; woh upar-neeche shift hue parallel copies hain. Ek path pin karne ke liye tumhe ek single anchor batana hota hai: ek ( x , y ) initial condition hai.
y ( x 0 ) = y 0 == x 0 == == y 0 == y ( x 0 ) = y 0 x 0 y 0 x 0 = 0 y 0 = 1 ( 0 , 1 )
Figure 3 dekho: bahut saari candidate curves slope rule share karti hain (pale chalk), lekin sirf woh jo marked anchor point (yellow dot) se guzarti hai woh humari solution hai.
h == h == x x 0 x 0 + h h h h = 0.1
x 1 y 1 == x 1 === x 0 + h == y 1 == y 1
∫ x 0 x 1 f d x ∫ sum hai. Yeh x 0 x 1 × height ka total change hai. Isliye parent likhta hai
y ( x 1 ) − y ( x 0 ) = ∫ x 0 x 1 f ( x , y ) d x . (end pe height) minus (start pe height) = beech ke saare chhote rises ka sum. Yeh exact hai — abhi tak koi approximation nahi.
Topic ko integral kyun chahiye? Kyunki slope rule sirf instantaneous directions deta hai; actual height recover karne ke liye tumhe un saari directions ko step mein accumulate karna hoga. Tiny pieces accumulate karna is integration. Euler aur Heun ke beech ka fark bas is ek sum ko estimate karne ke alag shortcuts hain — dekho Euler's Method (left-rectangle) versus Trapezoidal Rule (dono ends ka average).
k 1 k 2 == k 1 == start ka slope hai: k 1 = f ( x 0 , y 0 ) == k 2 == predicted end ka slope hai: k 2 = f ( x 1 , y 0 + h k 1 ) f
y 1 ( p ) ( p ) predicted hai — ek rough draft. y 1 ( p ) = y 0 + h k 1 k 2 y 1
Intuition Do slopes aur ek average kyunStart-slope k 1 k 1 k 2 2 1 ( k 1 + k 2 ) h
Derivative dy dx equals y prime
Slope rule dy dx equals f x y
Machine f x y outputs a slope
Initial condition y at x0 equals y0
Step size h and next point x1 y1
Two slopes k1 and k2 predictor
Heun method average of slopes
Khud test karo — right side cover karo.
x y x y
Machine f ( x , y ) Ek single number — position x y
Slope ko simple words mein define karo. Rise over run — jab tum thoda sa right step karo toh tum kitna upar jaate ho.
Curve pe slope = 0 Path momentarily flat hai (ek peak, valley, ya plateau).
y ′ d x d y Dono ek hi cheez hain — ek exact point pe y
d x d y = f ( x , y ) Path ka slope woh number hona chahiye jo rule f
Initial condition y ( x 0 ) = y 0 Slope rule akele infinitely many parallel curves allow karta hai; anchor ek sach wali path choose karta hai.
x 0 , y 0 , x 1 , h Start position, start height, next position = x 0 + h
∫ x 0 x 1 f d x Step mein height ka total change — saare tiny rises ka sum.
y 1 ( p ) Ek sasta Euler guess end height ka, sirf end-slope padhne ke liye banaya; correct karne ke baad discard ho jaata hai.
k 1 k 2 Start ka slope aur predicted end ka slope; Heun in dono ke average se chalta hai.
Euler's Method — predictor step aur §6 ka left-rectangle idea banata hai.Trapezoidal Rule — §6–§7 ka "dono ends ka average" integral shortcut.Order of Accuracy and Step Size — step size h Predictor-Corrector Methods — jahan k 1 , k 2 , y 1 ( p ) Runge-Kutta Methods — Heun as the two-slope RK2.Taylor Series Methods — derivative expansions accuracy kaise certify karte hain. 
