Foundations — Systems of first-order linear ODEs — matrix method

4.6.21 · D1 · Maths › Ordinary Differential Equations › Systems of first-order linear ODEs — matrix method

Yeh page parent note ke har symbol ko scratch se build karta hai. Upar se neeche padho; har block agle block ke liye notation earn karta hai. Parent dekho: Systems of first-order linear ODEs — matrix method.

1. Derivative kya hoti hai (prime mark ${}^{\prime }$)

Figure dekho: curve $y(t)$ ki value hai; amber seedhi line jo use bas chhu rahi hai woh tangent hai, aur uski steepness hi $y^{\prime }$ hai. Agar $y$ upar ja raha hai, $y^{\prime }>0$; flat matlab $y^{\prime }=0$; neeche girna matlab $y^{\prime }<0$.

2. Scalar equation $y^{\prime }=ay$ aur $e^{at}$

Yeh self-copying property exactly sentence $y^{\prime }=ay$ hai. Toh $y=C{e}^{at}$ isko solve karta hai, jahan $C$ starting value $y(0)$ hai.

3. Vectors — bold $\mathbf{x}$ aur uske components

Jab $\mathbf{x}$ time par depend karta hai, $\mathbf{x}(t)$ ek moving arrow hai — uski tip ek path trace karti hai. Tab ${\mathbf{x}}^{\prime }(t)$ ek aur arrow hai: velocity, jo dikhata hai ki tip kis taraf ja rahi hai aur kitni tezi se.

4. Matrices — letter $A$ aur $A\mathbf{v}$

$2\times 2$ example ke liye, $A\mathbf{v}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)=\left(\begin{array}{c}a{v}_1+b{v}_2\\ c{v}_1+d{v}_2\end{array}\right).$

Figure mein ek grey input arrow ek cyan output arrow ban jaata hai — aksar kahin aur point karta hua. Woh "kahin aur" hi saari mushkil hai: $A$ coordinates ko mix kar deta hai.

5. Eigenvectors aur eigenvalues — special directions

Ab §4 ka payoff. Zyaadaatar arrows $A$ se twist ho jaate hain. Lekin kuch khaas arrows same direction mein aate hain — bas lambe ya chote hote hain.

Figure mein amber arrow ek eigenvector hai: $A$ ke act karne ke baad, woh abhi bhi dashed line par hai — bas rescale hua hai. Grey arrow special nahi hai — woh apni line se hatt jaata hai.

6. Identity $I$, singularity, aur $\det$

Eigenvalues dhundhne ke liye hum $A\mathbf{v}=\lambda \mathbf{v}$ ko $(A-\lambda I)\mathbf{v}=0$ likhte hain. Teen symbols aate hain:

Equation $(A-\lambda I)\mathbf{v}=0$ ko ek nonzero $\mathbf{v}$ chahiye jo $0$ tak crush ho — yeh tab hota hai jab $\det (A-\lambda I)=0$. Yahi characteristic equation hai.

7. Complex numbers $i$, $\alpha \pm i\beta$, aur rotation

Kabhi kabhi characteristic equation ke real roots nahi hote — uske solutions mein $i$ aata hai.

Euler's formula (dekho Euler's Formula) ise rotation se jodta hai: $e^{(\alpha +i\beta )t}=e^{\alpha t}(\cos \beta t+i\sin \beta t).$ Ise yun padho: $e^{\alpha t}$ size set karta hai (badhta hai agar $\alpha >0$, decay karta hai agar $\alpha <0$), aur $\cos \beta t,\sin \beta t$ ise $\beta$ rate par circle ke around spin karte hain.

8. Constants $c_i$, superposition, aur initial conditions

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