Systems of first-order linear ODEs — matrix method

4.6.21 · Maths › Ordinary Differential Equations

KYA solve kar rahe hain

KAISE: solution scratch se derive karein

Step 1 — Ansatz substitute karo. Maano $x (t) = e^{λ t} v$ . Tab $x^{'} = λ e^{λ t} v, A x = e^{λ t} A v .$ Yeh step KYUN? Hum test kar rahe hain ki kya aisa self-similar solution exist kar sakta hai.

Step 2 — Scalar $e^{λ t} \neq = 0$ cancel karo. $λ e^{λ t} v = e^{λ t} A v ⟹ A v = λ v .$ Yeh step KYUN? $e^{λ t}$ kabhi zero nahi hota, isliye hum ise divide kar sakte hain — sirf ek purely algebraic condition bachti hai. Yahi bilkul eigenvalue equation hai.

Step 3 — Eigenvalues dhundho. $A v = λ v \Rightarrow (A - λ I) v = 0$ . Nonzero $v$ tab exist karta hai jab matrix singular ho: $det (A - λ I) = 0 (characteristic equation) .$ Yeh step KYUN? Ek homogeneous linear system $(A - λ I) v = 0$ ka nontrivial solution tab hi hota hai jab coefficient matrix invertible nahi ho, yaani determinant zero ho.

Step 4 — Superpose karo. System linear hai, isliye solutions ka koi bhi linear combination bhi solution hai (superposition principle). Agar $A$ ke paas $n$ independent eigenvectors $v_{1}, \dots, v_{n}$ hain jinka eigenvalues $λ_{1}, \dots, λ_{n}$ hain:

Complex aur repeated eigenvalues

Ek complex pair $λ = α + i β$ ke liye jiska eigenvector $v = p + i q$ ho, do real solutions hain:

\mathbf{x}_2=e^{\alpha t}(\mathbf p\sin\beta t+\mathbf q\cos\beta t).$$ > [!definition] Defective (repeated) case > Agar $\lambda$ ki algebraic multiplicity 2 ho lekin sirf ek eigenvector $\mathbf v$ ho, tab hume ek > **generalized eigenvector** $\mathbf w$ chahiye jo $(A-\lambda I)\mathbf w=\mathbf v$ solve kare. Tab ek doosra > independent solution hai $\mathbf{x}_2=e^{\lambda t}(t\,\mathbf v+\mathbf w)$. > *$t$ factor KYUN?* $e^{\lambda t}(t\mathbf v+\mathbf w)$ substitute karne par aur $(A-\lambda I)\mathbf w=\mathbf v$ use karne par > dono sides match ho jaate hain — $t$ woh missing degree of freedom provide karta hai. --- ![[4.6.21-Systems-of-first-order-linear-ODEs-—-matrix-method.png]] --- ## Worked Example 1 — distinct real eigenvalues $\mathbf{x}'=A\mathbf{x}$, $A=\begin{pmatrix}1&1\\4&1\end{pmatrix}$, $\mathbf x(0)=\binom{2}{-1}$ solve karo. **Step 1.** Characteristic equation: $\det(A-\lambda I)=(1-\lambda)^2-4=0$. *KYUN?* Nontrivial eigenvector ke liye $A-\lambda I$ singular chahiye. $(1-\lambda)^2=4 \Rightarrow 1-\lambda=\pm2 \Rightarrow \lambda=-1,\,3.$ **Step 2.** $\lambda=3$ ke liye eigenvector: $(A-3I)\mathbf v=\begin{pmatrix}-2&1\\4&-2\end{pmatrix}\mathbf v=0$ $\Rightarrow -2v_1+v_2=0\Rightarrow \mathbf v=\binom{1}{2}$. *KYUN?* Null direction ke liye singular system solve karo. $\lambda=-1$ ke liye: $\begin{pmatrix}2&1\\4&2\end{pmatrix}\mathbf v=0\Rightarrow 2v_1+v_2=0\Rightarrow\mathbf v=\binom{1}{-2}$. **Step 3.** General solution: $$\mathbf{x}=c_1e^{3t}\binom{1}{2}+c_2e^{-t}\binom{1}{-2}.$$ **Step 4 — IC apply karo.** $t=0$ par: $c_1\binom12+c_2\binom1{-2}=\binom{2}{-1}$. $c_1+c_2=2,\;2c_1-2c_2=-1\Rightarrow c_1=\tfrac34,\;c_2=\tfrac54.$ $$\boxed{\mathbf{x}(t)=\tfrac34e^{3t}\binom12+\tfrac54e^{-t}\binom1{-2}.}$$ --- ## Worked Example 2 — complex eigenvalues (spiral) $A=\begin{pmatrix}0&-2\\2&0\end{pmatrix}$. **Step 1.** $\det(A-\lambda I)=\lambda^2+4=0\Rightarrow\lambda=\pm2i$. Toh $\alpha=0,\beta=2$ — pure rotation. **Step 2.** $\lambda=2i$ ke liye eigenvector: $\begin{pmatrix}-2i&-2\\2&-2i\end{pmatrix}\mathbf v=0\Rightarrow -2i\,v_1-2v_2=0\Rightarrow v_2=-i\,v_1$. $\mathbf v=\binom{1}{-i}=\binom10+i\binom0{-1}$ lo, toh $\mathbf p=\binom10,\ \mathbf q=\binom0{-1}$. **Step 3.** Real solutions ($\alpha=0,\beta=2$): $$\mathbf{x}_1=\binom10\cos2t-\binom0{-1}\sin2t=\binom{\cos2t}{\sin2t},\quad \mathbf{x}_2=\binom{\sin 2t}{-\cos 2t}.$$ $$\mathbf{x}=c_1\binom{\cos2t}{\sin2t}+c_2\binom{\sin2t}{-\cos2t}.$$ *Circles KYUN?* $\alpha=0$ ka matlab growth nahi — trajectories closed circles hain (ek **center**). --- ## Worked Example 3 — repeated eigenvalue $A=\begin{pmatrix}3&1\\0&3\end{pmatrix}$. **Step 1.** $\det(A-\lambda I)=(3-\lambda)^2=0\Rightarrow\lambda=3$ (double). **Step 2.** $(A-3I)=\begin{pmatrix}0&1\\0&0\end{pmatrix}$. Eigenvector: $v_2=0\Rightarrow\mathbf v=\binom10$. Sirf **ek** eigenvector → defective. **Step 3.** Generalized eigenvector: $(A-3I)\mathbf w=\mathbf v\Rightarrow w_2=1$, $\mathbf w=\binom01$ lo. $$\mathbf{x}=c_1e^{3t}\binom10+c_2e^{3t}\!\left(t\binom10+\binom01\right).$$ *$t$ KYUN?* Yeh woh missing doosri direction manufacture karta hai jo ordinary exponentials nahi kar sakte the. --- > [!mistake] Steel-manned common errors > **(1) "Eigenvalues ki sankhya = independent solutions ki sankhya."** Sahi lagta hai kyunki *distinct* > eigenvalues ke liye yeh kaam karta hai. **Fix:** *eigenvectors* gino. Ek repeated eigenvalue bahut kam de sakta hai — tab generalized eigenvectors use karo. > > **(2) Solution mein eigenvector bhool jaana.** $\mathbf{x}=c_1e^{\lambda_1 t}+c_2e^{\lambda_2 t}$ > likhna (scalars). Scalar ODEs se tempting lagta hai. **Fix:** har term ke saath uska eigenvector $\mathbf v_i$ hona *zaroori* hai — solution ek vector hai. > > **(3) Dono complex conjugate pairs alag alag use karna AUR complex eigenvectors rakhna.** Yeh double-count karta hai. > **Fix:** ek complex pair $\alpha\pm i\beta$ exactly **do** real solutions deta hai (ek branch ke real aur imaginary parts). > > **(4) Determinant sign slip:** $\det(\lambda I-A)$ vs $\det(A-\lambda I)$ compute karna. Dono same > roots dete hain (yeh $(-1)^n$ se alag hote hain), lekin consistent raho taaki equation cleanly padhe. --- > [!recall]- Feynman: ek 12-saal ke bachche ko samjhao > Socho ek room mein ek swarm of arrows har particle ko push kar rahi hain. Normally pushes tangled hain. > Lekin kuch *special directions* hain jahan push sirf seedha bahar (ya seedha andar) usi line ke saath move karati hai, tez aur tez ya dheere aur dheere. Woh special lines eigenvectors hain; push kitni strongly kaam kare woh eigenvalue hai. Jab tumhe saari special lines pata hain, koi bhi motion unke saath move karne ka ek mix hai — toh messy swarm easy ho jaata hai! Agar ek special direction tumhe spin bhi karaati hai, toh tum spirals mein jaate ho. --- > [!mnemonic] "**E**very **V**ector **E**xponential **S**uperposes" > **E**igenvalues from $\det(A-\lambda I)=0$ → **V**ectors (eigenvectors) → **E**xponential $e^{\lambda t}$ har pair ke liye → **S**uperpose with constants $c_i$ → initial conditions se fix karo. --- ## Recall > [!recall] Quick self-test > - Woh ansatz jo $\mathbf{x}'=A\mathbf{x}$ decouple karta hai? $\;e^{\lambda t}\mathbf v$. > - Equation $A\mathbf v=\lambda\mathbf v$ solvable kab hoti hai? $\;\det(A-\lambda I)=0$. > - Complex pair mein real $\alpha<0$ ⇒ trajectory? $\;$ inward spiral (stable). ### #flashcards/maths System $\mathbf{x}'=A\mathbf{x}$ ko kaunsa ansatz solve karta hai? ::: $\mathbf{x}=e^{\lambda t}\mathbf v$ jisme $\mathbf v$ constant ho. Ansatz substitute karne ke baad kaunsi algebraic condition bachti hai? ::: $A\mathbf v=\lambda\mathbf v$, the eigenvalue equation. Eigenvalues kaise dhundhe jaate hain? ::: Characteristic equation $\det(A-\lambda I)=0$ solve karo. $e^{\lambda t}$ divide kyun kar sakte hain? ::: Kyunki exponential kabhi zero nahi hota, isliye relation constant factors ke liye hold karna chahiye. $n$ distinct eigenvalues ke liye general solution? ::: $\mathbf{x}=\sum_i c_i e^{\lambda_i t}\mathbf v_i$. Constants $c_i$ kya fix karta hai? ::: Initial condition $\mathbf{x}(0)=\mathbf x_0$. $\lambda=\alpha\pm i\beta$ ke liye $\alpha$ aur $\beta$ kya control karte hain? ::: $\alpha$: growth/decay rate; $\beta$: rotation frequency. $\lambda=\alpha+i\beta$, $\mathbf v=\mathbf p+i\mathbf q$ se do real solutions? ::: $e^{\alpha t}(\mathbf p\cos\beta t-\mathbf q\sin\beta t)$ aur $e^{\alpha t}(\mathbf p\sin\beta t+\mathbf q\cos\beta t)$. Repeated eigenvalue ka doosra solution $e^{\lambda t}(t\mathbf v+\mathbf w)$ kab hota hai? ::: Jab eigenvalue defective ho (eigenvectors uski multiplicity se kam hon); $\mathbf w$, $(A-\lambda I)\mathbf w=\mathbf v$ solve karta hai. $\mathbf{x}'=A\mathbf{x}$ ke liye stability rule? ::: Stable (decays) tab aur sirf tab jab saare eigenvalues ka real part negative ho. --- ## Connections - [[Eigenvalues and Eigenvectors]] — pure method ka algebraic engine. - [[Matrix Exponential]] — $\mathbf{x}(t)=e^{At}\mathbf x_0$ saare cases ko unify karta hai. - [[Diagonalization]] — distinct-eigenvalue case bas $A=PDP^{-1}$ change of basis hai. - [[Phase Portraits and Stability]] — eigenvalue signs se nodes, saddles, spirals, centers. - [[Second-Order Linear ODEs]] — ek $2\times2$ system mein reduce hota hai; same characteristic roots. - [[Euler's Formula]] — complex exponentials ko real oscillations mein convert karta hai. ## 🖼️ Concept Map ```mermaid flowchart TD SYS["System x' = Ax"] -->|vector version of| SCALAR["Scalar x' = ax, soln Ce^at"] SYS -->|A tangles variables| COUPLE["Coupled equations"] COUPLE -->|decouple via| ANSATZ["Ansatz x = e^lambda t v"] ANSATZ -->|substitute and cancel e^lambda t| EIG["Eigenvalue eqn Av = lambda v"] EIG -->|nontrivial v needs singular matrix| CHAR["det of A minus lambda I = 0"] CHAR -->|solve for| EVALS["Eigenvalues lambda_i"] EVALS -->|each gives| EVECS["Eigenvectors v_i"] EVECS -->|form| BASIS["Basic solutions e^lambda_i t v_i"] BASIS -->|superposition principle| GEN["General solution sum c_i e^lambda_i t v_i"] SCALAR -->|self-reproduces on differentiation| ANSATZ ```