Systems of ODEs

Definition

A system of n (1st-order) OSEs

functions $x_1,\dots ,x_n$ on $t$ with n initial conditions $x_1(t_0)=X_1,\dots ,x_n(t_0)=X_n$ has solutions $x_1={\varphi }_1(t),\dots ,x_n={\varphi }_n(t)$ on $(t_0-h,t_0+h)$ where $x_1^{\prime }=F_1(t,x_1,\dots ,x_n),\dots ,x_n^{\prime }=F_n(t,x_1,\dots ,x_n)$

Solution

Homogeneous linear systems differential equation with constant coefficients

Solution

We want to solve ${\mathbf{x}}^{\prime }=\mathbf{Ax}$. where $\mathbf{x}=\mathbf{x}(t)=\left(\begin{array}{c}{x}_1(t)\\ ⋮\\ x_n(t)\end{array}\right)$, $\mathbf{A}=\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{n1}& \dots & a_{nn}\end{array}\right)$

Try the form of the solution $\mathbf{x}$ for general n, $\mathbf{x}(t)=\xi e^{\lambda t}$, where $\xi$ is constant vector ${\mathbf{x}}^{\prime }=\lambda \xi e^{\lambda t}=\mathbf{A}\xi e^{\lambda t}\Rightarrow \mathbf{A}\xi =\lambda \xi$^[eigenvalue problem]

Real Eigenvalues ${\lambda }_1,\dots ,{\lambda }_n\to {\xi }^{(1)},\dots ,{\xi }^{(n)}$

Solutions: ${\mathbf{x}}^{(1)}={\xi }^{(1)}{e}^{{\lambda }_{1}t},\dots ,{\mathbf{x}}^{(n)}={\xi }^{(n)}{e}^{{\lambda }_{n}t}$

$W({\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)})=\left|\begin{array}{ccc}{\mathbf{x}}^{(1)}& \dots & {\mathbf{x}}^{(n)}\end{array}\right|=\left|\begin{array}{ccc}{\xi }^{(1)}& \dots & {\xi }^{(n)}\end{array}\right|e^{({\lambda }_1+\cdots +{\lambda }_n)t}\ne 0$ So, ${\xi }^{(1)},\dots ,{\xi }^{(n)}$ are linearly independent

General solution: $c_1{\xi }^{(1)}{e}^{{\lambda }_{1}t}+\cdots +c_n{\xi }^{(n)}{e}^{{\lambda }_{n}t},c_1,\dots ,c_n\in \mathbb{R}$

Complex Eigenvalues: ${\lambda }_1=\gamma +i\omega \to {\xi }^1,{\lambda }_2=\gamma -i\omega \to {\xi }^2$

Solutions: ${\xi }^{(1)}=\mathbf{a}+i\mathbf{b},{\xi }^{(2)}=\mathbf{a}-i\mathbf{b},\mathbf{a},\mathbf{b}$ are real vectors

${\mathbf{x}}^{(1)}={\xi }^{(1)}{e}^{{\lambda }_{1}t}=(\mathbf{a}+i\mathbf{b})e^{\gamma t}(\cos \omega t+i\sin \omega t)$ $=\underset{\mathbf{u}(t)}{\underbrace{e^{\gamma t}(\mathbf{a}\cos \omega t+\mathbf{b}\sin \omega t)}}+i\underset{\mathbf{v}(t)}{\underbrace{e^{\gamma t}(\mathbf{a}\sin \omega t+\mathbf{b}\cos \omega t)}}=\mathbf{u}(t)+i\mathbf{v}(t)$

$W(\mathbf{u},\mathbf{v})=\left|\begin{array}{cc}\mathbf{a}\cos \omega t+\mathbf{b}\sin \omega t& \mathbf{a}\sin \omega t+\mathbf{b}\cos \omega t\end{array}\right|e^{2\gamma t}$ $\Rightarrow W(\mathbf{u},\mathbf{v})(0)=\left|\begin{array}{cc}\mathbf{a}& \mathbf{b}\end{array}\right|\ne 0$

Repeated (real) root with less eigenvectors: ${\lambda }_1=\cdots ={\lambda }_n,{\xi }^{(1)},\dots ,{\xi }^{(k)},k<n$

Let ${\lambda }_1={\lambda }_2=\lambda ,\xi$ is only eigen vector, then ${\mathbf{x}}^{(1)}=\xi e^{\lambda t}$ (with $\mathbf{A}\xi =\lambda \xi$)

Try ${\mathbf{x}}^{(2)}=\xi t{e}^{\lambda t}+\eta e^{\lambda t}$, where generalized eigenvector $\eta$ is to be determined Then $({\mathbf{x}}^{(2)}{)}^{\prime }=\xi e^{\lambda t}+\xi \lambda t{e}^{\lambda t}+\eta \lambda e^{\lambda t}=\mathbf{A}{\mathbf{x}}^{(2)}=\mathbf{A}(\xi t{e}^{\lambda t}+\eta e^{\lambda t})$ $\Rightarrow (\mathbf{A}-\lambda I)\eta =\xi \Rightarrow (\mathbf{A}-\lambda I{)}^2\eta =\underset{0}{\underbrace{\mathbf{A}\xi -\lambda I\xi }}\Rightarrow (\mathbf{A}-\lambda I{)}^2=0$

Triple roots: ${\mathbf{x}}^{(3)}=\xi \frac{t^2}{2}{e}^{\lambda t}+\eta t{e}^{\lambda t}+\zeta e^{\lambda t}$, where $\zeta$ is to be determined

Link to original

Facts

Solution of systems of ODEs

Let x_i' = F_i(t, x_1, \dots, x_n), \quad _{i=1, \dots, n} be a system of n(1th-order) ODEs.

If ${\varphi }_1(t){,}_{i=1,\dots ,n}$ is defined and differentiable on $I$ and $x_i^{\prime }={\varphi }_i^{\prime }(t)=F_i(t,{\varphi }_1(t),\dots ,{\varphi }_n(t))\forall t\in I$ we say that $\{x_1={\varphi }_1(t),\dots ,x_n={\varphi }_n(t)\}$ is a solution on an open interval $I=(\alpha ,\beta )$

Conversion of an ODE

An n-th order ODE $y^{(n)}=F(t,y,y^{\prime },\dots ,y^{(n-1)})$ can be converted to a system of 1st-order ODEs by setting $x_1=y,x_2=y^{\prime },\dots ,x_n=y^{(n-1)}$

This is a system of the form $x_1^{\prime }=x_2,x_2^{\prime }=x_3,\dots ,x_{n-1}^{\prime }=x_n,x_n^{\prime }=F(t,x_1,\dots ,x_n)$

Linear system

If $F_1,\dots ,F_n$ are linear on $x_1,\dots ,x_n$, a system of ODEs is called linear

If a system of 1st-linear ODEs is linear,
then $\exists p_{ij}(t),g_i(t)$, s.t. $x_i^{\prime }=\sum _{j=1}^n{p}_{ij}(t)x_j+g_i(t)$ or $\mathbf{x}=\mathbf{Px}+\mathbf{g}(t)$
where $\mathbf{x}(t)=\left[\begin{array}{c}{x}_1(t)\\ ⋮\\ x_n(t)\end{array}\right],\mathbf{P}(t)={\left[\begin{array}{c}{P}_{ij}(t)\end{array}\right]}_{1\le i,j\le n},\mathbf{g}(t)=\left[\begin{array}{c}{g}_1(t)\\ ⋮\\ g_n(t)\end{array}\right]$

If $g_1(t),\dots ,g_n(t)=0$, the system is called homogeneous
If $g_i(t)\ne 0$, the system is called non-homogeneous

Existence and Uniqueness Theorem

Let x_i' = F_i(t, x_1, \dots, x_n),\quad _{i=1, \dots, n} be a system of (1st-order) ODEs, with $x_i(t_0)=x_i^0\in \mathbb{R}$

If $F_i,\frac{\partial F_i}{\partial x_j}{,}_{1\le i,j\le n}$ are continuous in $R\subseteq t,x_1,x_2,\dots ,x_n$ space by $R=\{(t,x_1,\dots ,x_n)|\alpha <t<\beta \}$ and if $(t_0,x_1^0,\dots ,x_n^0)\in R$
then, $\exists h>0$ and a unique solution $x_1={\varphi }_1(t),\dots ,x_n={\varphi }_n(t)$ defined on $(t_0-h,t_0+h)$ satisfying I.V.P

Existence and Uniqueness Theorem for Linear ODE

Let x_i' = F_i(t, x_1, \dots, x_n),\quad _{i=1, \dots, n} be a system of (1st-order) linear ODEs

If P_{i, j}, g_i,\quad _{1 \le i, j \le n} are continuous on $I=\{t|\alpha <t<\beta \}$ and $x_i(t_0)=x_i^0\in \mathbb{R},t_0\in I$
then, $\exists$ a unique solution $x_1={\varphi }_1(t),\dots ,x_n={\varphi }_n(t)$ defined on $I=(\alpha ,\beta )$ satisfying I.V.P

Basic Theory of Systems of ODEs

If ${\mathbf{x}}^{(i)}$ is solution of ${\mathbf{x}}^{\prime }=P(t)\mathbf{x}$, then $\forall c_i\in \mathbb{R},\sum _{i=1}^n{c}_i{\mathbf{x}}^{(i)}$ is a solution

Let ${\mathbf{x}}^{\prime }=P(t)\mathbf{x}$ be a homogeneous linear system of n ODEs on $I$.

A basis (or fundamental solution) of solutions of the system is a linearly independent set of n solutions

If $\{{\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}\}$ is a basis, $\mathbf{x}=c_1{\mathbf{x}}^{(1)}+\cdots +c_n{\mathbf{x}}^{(n)}$ is called a general solution

Let ${\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}$ be $n\times 1$ vectors (of functions) They are linearly independent $\iff$ if $c_1,\dots ,c_n\in \mathbb{R},\sum _{i=1}^n{c}_i{\mathbf{x}}^{(i)}=0$ then $c_1=\cdots =c_n=0$ They are linearly dependent $\iff \exists c_1,\dots ,c_n$, not all zero, s.t. $\sum _{i=1}^n{c}_i{\mathbf{x}}^{(i)}=0$ They are linearly independent at a point $t_0\iff {\mathbf{x}}^{(1)}(t_0),\dots ,{\mathbf{x}}^{(n)}(t_0)$ $W({\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}):=\left|\begin{array}{ccc}{\mathbf{x}}^{(1)}& \dots & {\mathbf{x}}^{(n)}\end{array}\right|=|\mathbf{X}(t)|$, where $|\mathbf{X}(t)|$ is the Wronskian Determinant of ${\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}$

Wronskian and systems of ODEs

If n-th order linear ODE and the corresponding system of linear ODEs is given,
then they have the same Wronskian Determinant

Let $t_0\in I$. $W({\mathbf{x}}^{(1)}(t_0),\dots ,{\mathbf{x}}^{(n)}(t_0))\ne 0\iff {\mathbf{x}}^{(1)}(t_0),\dots ,{\mathbf{x}}^{(n)}(t_0)$ are linearly independent

If $\{{\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}\}$ is a basis for the solution of a homogeneous linear system on $I$,
then $\forall$ solution $\varphi$, $\exists c_1,\dots ,c_n\in \mathbb{R}$ s.t. $\varphi (t)=\sum _{i=1}^n{c}_i{\mathbf{x}}^{(i)}$

${\mathbf{x}}^{(1)}(t),\dots ,{\mathbf{x}}^{(n)}(t_0)$ are linearly independent on $I=(\alpha ,\beta )$ $\Rightarrow {\mathbf{x}}^{(1)}(t_0),\dots ,{\mathbf{x}}^{(n)}(t_0)$ are linearly independent

So, ${\mathbf{x}}^{(1)}(t),\dots ,{\mathbf{x}}^{(n)}(t_0)$ are linearly independent at some $t=t_0$, then ${\mathbf{x}}^{(1)}(t),\dots ,{\mathbf{x}}^{(n)}(t_0)$ are linearly independent.

If ${\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}$ are linear solutions of homogeneous linear ODEs on $x_1,\dots ,x_n$ on $I=(\alpha ,\beta )$, then $W({\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)})\ne 0\forall t\in I\iff W({\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)})(t_0)\ne 0$ for some $t_0\in I$

Given a linear system of ODE on $I=(\alpha ,\beta )$, $\exists$ a basis $\{{\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}\}$