Differential Equations — Foundations¶

Job: Classify ODEs/PDEs; state local existence–uniqueness for IVPs; frame linear theory as spectral / fundamental-matrix problems; orient stability and boundary-value structure used in dynamics and PDE.

Core¶

Axis	Distinction
ODE / PDE	one independent variable / several
Order	maximal derivative order
Linear / nonlinear	affine in $(y,y',\ldots,y^{(n)})$ vs not
Autonomous	$f$ independent of $t$ for $\dot x=f(x)$
IVP / BVP	initial data at a point / conditions on a boundary

Normal form IVP: $\dot x=f(t,x)$, $x(t_0)=x_0\in\mathbb{R}^n$ (or Banach space).

Local existence and uniqueness¶

Picard–Lindelöf. If $f$ continuous on an open set $U\subset\mathbb{R}\times\mathbb{R}^n$ and locally Lipschitz in $x$ uniformly in $t$ on compact subsets, then for each $(t_0,x_0)\in U$ there is a unique maximal solution $x\in C^1(J)$ on an open interval $J\ni t_0$. Proof: contraction mapping on the integral operator $$ (Tx)(t)=x_0+\int_{t_0}^t f(s,x(s))\,ds $$ in a closed ball of $C([t_0-\delta,t_0+\delta])$ with weighted supremum norm.

Peano: continuity of $f$ alone $\Rightarrow$ local existence (no uniqueness). Osgood / one-sided Lipschitz refine uniqueness criteria. Carathéodory: $f$ measurable in $t$, continuous in $x$, integrably bounded $\Rightarrow$ absolutely continuous solutions of the integral equation.

Escape-time / maximal interval. On a maximal interval $(T_-,T_+)$, if $T_+<\infty$ then the graph leaves every compact subset of $U$ (blow-up or approach $\partial U$). Linear growth $\|f(t,x)\|\le A(t)(1+\|x\|)$ with $A\in L^1_{\mathrm{loc}}$ $\Rightarrow$ global existence.

Linear systems¶

Time-varying: $\dot x=A(t)x+g(t)$ with $A\in L^1_{\mathrm{loc}}$, $g\in L^1_{\mathrm{loc}}$. Solutions form an affine space of dimension $n$; fundamental matrix $\Phi(t)$ satisfies $\dot\Phi=A(t)\Phi$, $\Phi(t_0)=I$. Liouville–Ostrogradski: $$ \det\Phi(t)=\det\Phi(t_0)\exp\Bigl(\int_{t_0}^t\operatorname{tr}A(s)\,ds\Bigr). $$ Variation of parameters: $$ x(t)=\Phi(t)x_0+\Phi(t)\int_{t_0}^t\Phi(s)^{-1}g(s)\,ds. $$

Constant coefficients: $\Phi(t)=e^{tA}$; spectral / Jordan decomposition of $A$ yields modes $t^k e^{\lambda t}v$. Stability of $0$ for $\dot x=Ax$: asymptotic stability $\Leftrightarrow$ $\operatorname{Re}\sigma(A)<0$; Lyapunov stability $\Leftrightarrow$ no positive real part and Jordan blocks for pure imaginary eigenvalues are trivial.

Floquet (periodic $A(t+T)=A(t)$): $\Phi(t)=P(t)e^{tB}$ with $P$ $T$-periodic; stability from eigenvalues of the monodromy matrix $\Phi(T)$.

Nonlinear structure¶

First integrals / reduction: autonomous planar systems; exact equations $M\,dx+N\,dy=0$ with $\partial M/\partial y=\partial N/\partial x$; integrating factors; Noether symmetries for variational ODEs.

Gronwall inequality. If $u(t)\le a+\int_{t_0}^t b(s)u(s)\,ds$ with $b\ge0$, then $u(t)\le a\exp\int_{t_0}^t b$. Controls continuous dependence on data and parameters: $$ |x(t;x_0)-x(t;y_0)|\le e^{L|t-t_0|}|x_0-y_0| $$ under uniform Lipschitz constant $L$.

Hartman–Grobman. If $f(0)=0$ and $Df(0)$ has no eigenvalue with $\operatorname{Re}\lambda=0$, then near $0$ the flow of $\dot x=f(x)$ is topologically conjugate to the linear flow of $\dot x=Df(0)x$.

Lyapunov functions. $V$ positive definite, $\dot V\le0$ $\Rightarrow$ stability; $\dot V$ negative definite $\Rightarrow$ asymptotic stability (LaSalle invariance for the semidefinite case).

Center manifold / normal forms (pointer). When hyperbolic assumptions fail, reduce to center manifold; Poincaré–Dulac normal forms organize resonant nonlinear terms.

Sturm–Liouville and BVPs¶

Regular SL problem $-(py')'+qy=\lambda wy$ with separated BC: eigenvalues real, simple, eigenfunctions orthogonal in $L^2(w)$; oscillation theorems count zeros. Nonself-adjoint and singular SL (Weyl $L^2$ classification) feed spectral theory of 1D Schrödinger operators.

Fredholm alternative for linear BVP $Lx=g$ on a compact interval: solvability $\Leftrightarrow$ $g\perp\ker L^*$; resonant forcing produces secular terms in time-periodic problems.

PDE classification (orientation)¶

Second-order linear $a^{ij}u_{x_ix_j}+\cdots=0$: type by signature of $(a^{ij})$ — elliptic / parabolic / hyperbolic (model equations $\Delta u=0$, $u_t=\Delta u$, $u_{tt}=\Delta u$). Well-posedness packages differ radically by type; see PDE foundations.

See it¶

$A=\begin{pmatrix}0&1\\-\omega^2&0\end{pmatrix}$: eigenvalues $\pm i\omega$, solutions $\mathrm{span}\{\cos\omega t,\sin\omega t\}$ for the harmonic oscillator $y''+\omega^2 y=0$ written as a first-order system. Resonant forcing $g(t)=e^{i\omega t}$ produces secular terms $t e^{i\omega t}$ unless projected orthogonal to the kernel of the adjoint (Fredholm alternative for periodic problems).

Comparison: logistic vs finite-time blow-up. $\dot x=x(1-x)$ has global bounded solutions for $x_0>0$; $\dot x=x^2$ blows up at $T=1/x_0$. Same polynomial degree, different growth structure — Lipschitz on compacts never forbids blow-up at infinity of the domain.

Check¶

Does $y'=\sqrt{|y|}$, $y(0)=0$ have a unique $C^1$ solution?

No. $y\equiv0$ and the family $y(t)=0$ for $t\le a$, $y(t)=\frac14(t-a)^2$ for $t>a$ ($a\ge0$) are solutions. $f$ is continuous but not Lipschitz at $0$.

State the precise local uniqueness hypothesis in Picard–Lindelöf.

Local Lipschitz continuity of $f$ in the state variable (uniformly in $t$ on compact time intervals), not mere continuity. Continuity alone is Peano (existence only).

When is the zero solution of $\dot x=Ax$ asymptotically stable?

Iff every eigenvalue of $A$ has strictly negative real part. Pure imaginary eigenvalues with nontrivial Jordan blocks already destroy even Lyapunov stability.

Pitfalls¶

Global existence $\neq$ uniqueness of the maximal interval endpoint: escape to infinity or leave the domain of $f$.
Linear constant-coeff recipes fail for variable $A(t)$ without Floquet / fundamental matrix.
Boundary-value problems are not IVPs: eigenvalues of Sturm–Liouville appear; uniqueness can fail on resonant intervals.
Linearization is not enough at nonhyperbolic equilibria (center manifolds; Hopf bifurcation).
Distributional / weak solutions for PDE are a different package from classical ODE IVPs.

Refs¶

Coddington–Levinson, Theory of Ordinary Differential Equations
Hale, Ordinary Differential Equations
Hartman, Ordinary Differential Equations
Arnold, Ordinary Differential Equations
Teschl, Ordinary Differential Equations and Dynamical Systems
Chicone, Ordinary Differential Equations with Applications

Next¶

First-order ODEs · Second-order ODEs · Laplace transforms · Dynamical systems · PDE · Linear algebra / eigenvalues