Foundations — Solving ODEs with Laplace (including discontinuous forcing)

4.6.30 · D1 · Maths › Ordinary Differential Equations › Solving ODEs with Laplace (including discontinuous forcing)

Pehle aap parent note ka ek bhi worked example padh sako, usse pehle page par har ek mark aapko khud se pata hona chahiye. Neeche, har symbol ko kuch nahi se build kiya gaya hai: pehle plain words mein, phir uski picture, phir kyun topic yahan aage nahi badh sakta uske bina. Upar se neeche padho — har item upar wale par lean karta hai.

1. Time variable $t$ aur ek function $f(t)$

Picture. $t$ ke liye ek horizontal axis aur value ke liye ek vertical axis draw karo. Jab aapki pencil daayein chalti hai (time guzarta hai), curve upar-neeche hoti hai. Woh wiggling curve hi $f$ hai.

Topic ko yeh kyun chahiye. Ek ODE mein har unknown aisi hi time ki ek function hoti hai — position, voltage, temperature. Poora game yeh hai: curve $y(t)$ dhundho.

2. Derivative $f^{\prime }(t)$ — curve ki slope

Picture. Curve ke ek point par ek chhoti seedhi ruler rakh do taaki woh sirf use kiss kare (tangent line). Uski tilt wahan $f^{\prime }$ hai. Tilt upar-daayein → $f^{\prime }>0$; flat → $f^{\prime }=0$; neeche-daayein → $f^{\prime }<0$.

Topic ko yeh kyun chahiye. Ek differential equation woh equation hai jo $y$ ko uske apne slopes $y^{\prime }$, $y^{\prime \prime }$ se jorti hai. Yahi self-reference hai jo ise mushkil banata hai — aur jise Laplace suljhaega. Hum jo equations solve karte hain unki family dekhne ke liye Linear Constant-Coefficient ODEs dekho.

3. Initial conditions $f(0)$ aur $f^{\prime }(0)$

Picture. Apne $t$–value graph par, $f(0)$ wahan hai jahan curve vertical axis ko cross karti hai; $f^{\prime }(0)$ us crossing point par curve ki tilt hai.

Topic ko yeh kyun chahiye. Ek ODE akele infinitely many solution curves (ek poora family) rakhta hai. ICs woh ek curve chunti hain jo aap chahte ho. Parent note ki headline feature — "Laplace initial conditions khud hi kha jaata hai" — tab tak kuch matlab nahi rakhti jab tak aap jaante nahi ki $f(0),f^{\prime }(0)$ HAIN kya.

4. Exponential $e^{-st}$ — ek decaying weight

Picture. Ek ski-slope jo left par height $1$ par pinned hai, right par zameen ki taraf sink ho rahi hai. Bada $s$ = teekhi dive.

Topic ko yeh kyun chahiye. $e^{-st}$ Laplace integral ke andar multiplier hai. Koi aur weight derivative-to-multiplication ka itna clean trade nahi deta.

5. Integral ${\int }_0^{\infty }$ — total accumulated area

Picture. Curve ke neeche ki region ko infinitely patli vertical strips mein kaato; unki saari areas jodo. Tall symbol $\int$ ek stretched "S" hai "sum" ke liye.

Topic ko yeh kyun chahiye. Laplace transform EK bada integral hai. Aur poora trick "differentiation becomes multiplication by $s$" integration by parts se prove hota hai — ek area-rearranging rule — isliye aapko comfortable hona chahiye ki ek integral area collect karta hai.

6. Transform $\mathcal{L}$ aur $s$-domain, $F(s)$

Picture. Do side-by-side worlds. Left: time ($t$) mein wiggling curve. Right: $s$-land mein ek alag, aam taur par simpler curve. Machine $\mathcal{L}$ ek arrow hai jo aapko left → right le jaati hai; ek return arrow ${\mathcal{L}}^{-1}$ aapko wapas laata hai.

Topic ko yeh kyun chahiye. Yahi poori strategy hai: hard $t$-land chhodo, $s$-land mein easy algebra karo, ghar wapas aao. Upar ke har symbol ($t$, $f$, $e^{-st}$, $\int$) sirf is ek line ko build karne ke liye exist karte hain. Integral kyun converge karta hai ke deeper reasons Laplace Transform — Definition and Existence mein hain.

7. Unknown $Y(s)$ aur inversion ${\mathcal{L}}^{-1}$

Picture. §6 jaise hi two-world picture, lekin ab aap right par start karte ho (aapne $Y(s)$ algebra se dhundha) aur return arrow par left ki taraf sawari karte ho taaki answer curve $y(t)$ reveal ho.

Topic ko yeh kyun chahiye. Recipe ka Step 3 hai "invert." Return trip ke bina, $s$-land ke answers bekar hain.

8. Partial fractions — un-mixing tool

Picture. Left par ek heavy fraction, ek "=" sign, aur right par do light lego-brick fractions jo aap individually dekh sakte ho.

Topic ko yeh kyun chahiye. Algebra ke baad, $Y(s)$ ek messy fraction hai. Return machine ${\mathcal{L}}^{-1}$ sirf simple bricks jaanti hai. Partial fractions mess ko brick-sized pieces mein kaat deta hai. Full mechanics Partial Fractions mein hain.

9. Step $u(t-a)$ aur impulse $\delta (t-a)$ — switches aur hammer-blows

Picture. Step ek flat floor hai jo $t=a$ par achanak ek seetha upar chadh jaata hai. Impulse ek single vertical arrow hai jo $t=a$ par khada hai.

Topic ko yeh kyun chahiye. Real inputs kisi time par on hote hain ya ek baar strike karte hain. Yeh do objects "off, phir on" aur "ek akela kick" ke liye clean algebra dete hain — discontinuous forcing jiske liye parent topic mashur hai. Inki careful definitions Heaviside Step and Dirac Delta Functions mein hain.

Yeh topic ko kaise feed karte hain

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.