MATH383: Linear Systems of Differential Equations

Overview

• Motivating examples:

  • Screening clinics

  • SIR, Susceptible, Infectious, Recovered (NLSDE)

  • Coupled oscillators

• Formulation of linear systems of differential equations (LSDE)

• Homogeneous linear systems of differential equations (hLSDE)

Linear systems of differential systems

• Dependent variables $𝒚:ℝ\to {ℝ}^n$

  ${\displaystyle \left(\mathrm{IVP}\right)𝒚\left(t\right)=\left(\begin{array}{l}{y}_1\left(t\right)\\ ⋮\\ y_n\left(t\right)\end{array}\right),{𝒚}^{\text{'}}=𝑨\left(t\right)𝒚+𝒇\left(t\right)=𝑭(t,𝒚),𝒚\left(0\right)=𝒌={𝒚}_0.}$

Theorem. If $𝑨:ℝ\to {ℝ}^{n\times n}$ and $𝒇:ℝ\to {ℝ}^n$ are continuous on $(a,b)$ that contains the initial condition, then the initial value problem (IVP) has a unique solution on $(a,b)$.

• $z^{\left(n\right)}+p_1{z}^{(n-1)}+\cdots +p_{n-1}{z}^{\text{'}}+p_{n}z=-g\left(t\right)$

  ${\displaystyle z=y_1,z^{\text{'}}=y_2=y_1^{\text{'}},\dots ,z^{(n-1)}=y_n=y_{n-1}^{\text{'}}}$ ${\displaystyle y_n^{\text{'}}=-g-p_1{y}_n-\dots -p_n{y}_1.}$

Homogeneous systems of linear differential equations

• ${𝒚}^{\text{'}}=𝑨𝒚$ is a homogeneous LSDE (hLSDE). $𝒚=𝟎$ is a trivial solution.

• $𝒮=\{{𝒚}_1,{𝒚}_2,\dots ,{𝒚}_n\}.$ Linear combination

  ${\displaystyle \left(\mathrm{LC}\right)𝒚=c_1{𝒚}_1+c_2{𝒚}_2+\dots +c_n{𝒚}_n.}$

• If any solution of ${𝒚}^{\text{'}}=𝑨𝒚$ can be written as the linear combination (LC), then $𝒮$ is a fundamental set of solutions.

Theorem. The set of solutions $𝒮=\{{𝒚}_1,{𝒚}_2,\dots ,{𝒚}_n\}$ is a fundamental set if and only if $𝒮$ is linearly independent.