Existence and uniqueness for systems

The existence and uniqueness theorem for systems generalizes the scalar Existence and uniqueness theorem to vector ODEs ${\mathbf{x}}^{\prime }=\mathbf{F}(t,\mathbf{x})$, $\mathbf{x}(t_0)={\mathbf{x}}_0$.

The conditions are essentially the same, lifted to vectors and partial derivatives:

(EU1) The functions $F_i$ and $\frac{\partial F_i}{\partial x_j}$ for all $i,j=1,\dots ,n$ are continuous in some “$(n+1)$-dimensional rectangle”

$R=\{(t,x_1,\dots ,x_n):\alpha <t<\beta ,{\alpha }_k<x_k<{\beta }_k\text{ for }k=1,\dots ,n\}\subset {\mathbb{R}}^{n+1}$

(EU2) $(t_0,{\mathbf{x}}_0)\in R$.

Then there exists $h>0$ and a unique solution $\mathbf{x}(t)$, defined on $(t_0-h,t_0+h)$, of the IVP.

Why all those continuities

For a vector ODE, $\mathbf{F}:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^n$ has $n$ scalar component functions $F_i$. Each component depends on $t$ and the $n$ unknowns $x_1,\dots ,x_n$. So we need to check:

• Each $F_i$ is continuous (continuity of the right-hand side).
• Each partial derivative $\partial F_i/\partial x_j$ is continuous (smoothness of the right-hand side as a function of the state vector).

The Jacobian matrix $\partial F_i/\partial x_j$ is the multi-dimensional analog of $\partial f/\partial y$ in the scalar theorem. Its continuity ensures uniqueness.

Linear systems: stronger result

For linear systems ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}+\mathbf{b}(t)$, a much stronger result holds.

Theorem (existence and uniqueness for linear systems): if the entries $a_{ij}(t)$ and $b_i(t)$ are continuous on an open interval $I=(\alpha ,\beta )$ and $t_0\in I$, then there exists a unique solution $\mathbf{x}(t)$ to the IVP

${\mathbf{x}}^{\prime }=A(t)\mathbf{x}+\mathbf{b}(t),\mathbf{x}(t_0)={\mathbf{x}}_0$

for any choice of ${\mathbf{x}}_0\in {\mathbb{R}}^n$. Moreover, the solution is defined on all of $I$ (global, not just local).

This is the same upgrade as in the scalar case: linear systems’ solutions are guaranteed to exist on the entire interval where the coefficients are continuous, not just locally.

Why linearity helps

For nonlinear systems, solutions can “blow up” in finite time, even when $\mathbf{F}$ is smooth. Example: $x^{\prime }=x^2$ with $x(0)=1$ has solution $x(t)=1/(1-t)$ — infinite at $t=1$. So even though the right-hand side is smooth everywhere, the solution doesn’t exist past $t=1$.

Linear systems can’t do this. The bound $|x(t)|\le |{\mathbf{x}}_0|e^{Mt}$ (where $M$ depends on bounds of $A$) keeps solutions finite over any finite interval.

Affine functions

A function $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is affine if it has the form $f(\mathbf{x})=A\mathbf{x}+\mathbf{b}$ for a constant matrix $A$ and constant vector $\mathbf{b}$. Affine = linear + translation.

A linear system of ODEs ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}+\mathbf{b}(t)$ has an affine right-hand side at each fixed $t$ — linear in $\mathbf{x}$ plus a translation.

This terminology shows up in the formal definitions but in practice doesn’t change the analysis.

Superposition for systems

The Superposition principle extends to systems. If $\mathbf{y},\mathbf{z}$ are solutions of the homogeneous system ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}$, then for any constants $c_1,c_2$:

$\mathbf{x}(t)=c_1\mathbf{y}(t)+c_2\mathbf{z}(t)$

is also a solution. The solution space is a vector space.

In context

The theorem provides the framework for solving $n$-dimensional ODE systems — guaranteeing that the solution methods you apply yield well-defined, unique solutions. For the practical solution methods, see System of first-order linear ODEs and the case-by-case eigenvalue analyses (Distinct real eigenvalues case, Complex conjugate eigenvalues case, Repeated eigenvalues case).

For when you have multiple solutions and want to know if they form a basis, see Linear independence of vector functions and the matrix-form Wronskian.