3 Fundamental Forms and Classification
An ODE of order p can be written as
F(t,y,y',\ldots,y^{(p)})=0, \qquad y^{(p)}=f(t,y,\ldots,y^{(p-1)}).
Setting x=(y,y',\ldots,y^{(p-1)})^T gives a first-order system
x'=f(t,x),\qquad x\in\mathbb{R}^n.
The initial-value problem (IVP), or Cauchy problem, is
\boxed{x'=f(t,x),\qquad x(t_0)=x_0.}
An autonomous system has the form x'=f(x); a nonautonomous system has the form x'=f(t,x). A linear system is
x'=A(t)x+b(t),
and is homogeneous when b=0 and has constant coefficients when A(t)=A.
3.1 Important Scalar ODEs
A separable equation satisfies
y'=a(t)b(y),\qquad \int\frac{dy}{b(y)}=\int a(t)\,dt+C.
Zeros of b may give constant solutions. For a linear equation,
y'+a(t)y=b(t),\qquad \mu(t)=\exp\!\left(\int a(t)\,dt\right),
the integrating-factor formula is
y(t)=\mu(t)^{-1}\left[C+\int\mu(t)b(t)\,dt\right].
A Bernoulli equation and its standard substitution are
y'+a(t)y=b(t)y^\alpha,\quad \alpha\ne0,1, \qquad z=y^{1-\alpha}.
A Riccati equation is
y'=a(t)+b(t)y+c(t)y^2.
Given a particular solution y_p, the substitution y=y_p+1/z produces a linear equation. An exact equation satisfies
M(t,y)\,dt+N(t,y)\,dy=0,\qquad M_y=N_t,
and locally has the form \Phi(t,y)=C, with \Phi_t=M and \Phi_y=N.
4 Existence, Uniqueness, and Continuation
A classical solution x:I\to\mathbb{R}^n is continuously differentiable and satisfies the ODE pointwise. Its integral form is
x(t)=x_0+\int_{t_0}^t f(s,x(s))\,ds.
4.1 Peano’s Theorem
Continuity of f implies local existence, but not necessarily uniqueness.
4.2 Picard–Lindelöf Theorem
If f is continuous in t and locally Lipschitz in x, the IVP has a unique local solution. The Lipschitz condition is
\|f(t,x)-f(t,y)\|\le L\|x-y\|.
If f is C^1 with respect to x, it is locally Lipschitz in x. Picard iteration is
x^{(k+1)}(t)=x_0+\int_{t_0}^t f(s,x^{(k)}(s))\,ds.
4.3 Carathéodory Solutions
Carathéodory theory permits measurable dependence on t and continuous dependence on x, together with suitable bounds. Solutions are absolutely continuous and satisfy the ODE almost everywhere.
4.4 Maximal Interval
The unique solution extends to a maximal interval (\alpha,\beta). It can be continued as long as its graph remains in a compact subset of the domain of f. At-most-linear growth,
\|f(t,x)\|\le a(t)+b(t)\|x\|,
provides standard conditions for global existence.
5 Dependence on Data and Grönwall’s Inequality
If f is Lipschitz in x, then
\|x(t)-y(t)\|\le \mathrm{e}^{L|t-t_0|}\|x(t_0)-y(t_0)\|.
Grönwall’s inequality states that
u(t)\le a+\int_{t_0}^t b(s)u(s)\,ds \quad\Longrightarrow\quad u(t)\le a\exp\!\left(\int_{t_0}^t b(s)\,ds\right).
For x'=f(t,x,\mu), sufficient regularity gives continuous or differentiable dependence of the solution on initial data and parameters.
6 Flows, Variational Equations, and Sensitivities
For x'=f(x), the flow satisfies
\varphi_0=I,\qquad \varphi_{t+s}(x)=\varphi_t(\varphi_s(x)).
For nonautonomous systems one uses \Phi(t,s,x) with the corresponding composition property. Linearization along x(t) gives
\delta x'=\nabla_x f(t,x(t))\,\delta x.
The sensitivity with respect to the initial condition satisfies
S=\frac{\partial x(t;t_0,x_0)}{\partial x_0},\qquad S'=\nabla_x f\,S,\quad S(t_0)=I.
Parameter sensitivity satisfies
S_\mu'=\nabla_xf\,S_\mu+f_\mu.
7 Nonautonomous Linear Systems
For
x'=A(t)x+b(t),
a fundamental matrix satisfies
\Phi'=A(t)\Phi,\qquad \det\Phi\ne0.
The state-transition matrix obeys
\Phi(s,s)=I,\qquad \Phi(t,r)\Phi(r,s)=\Phi(t,s),\qquad \Phi(t,s)^{-1}=\Phi(s,t).
Variation of constants gives
\boxed{x(t)=\Phi(t,t_0)x_0+ \int_{t_0}^t\Phi(t,s)b(s)\,ds.}
The Wronskian is
W(t)=\det[x_1(t)\ \cdots\ x_n(t)],
and Liouville’s formula is
\det\Phi(t)=\det\Phi(t_0) \exp\!\left(\int_{t_0}^t\operatorname{tr}A(s)\,ds\right).
8 Constant-Coefficient Linear Systems
For x'=Ax,
x(t)=\mathrm{e}^{A(t-t_0)}x_0.
The matrix exponential satisfies
\mathrm{e}^{At}=\sum_{k=0}^\infty\frac{t^kA^k}{k!},\qquad \frac{d}{dt}\mathrm{e}^{At}=A\mathrm{e}^{At},\qquad (\mathrm{e}^{At})^{-1}=\mathrm{e}^{-At},
\det(\mathrm{e}^{At})=\mathrm{e}^{t\operatorname{tr}A}.
If AB=BA, then \mathrm{e}^{A+B}=\mathrm{e}^A\mathrm{e}^B. If A=V\Lambda V^{-1}, then
\mathrm{e}^{At}=V\mathrm{e}^{\Lambda t}V^{-1}.
For J=\lambda I+N with N^p=0,
\mathrm{e}^{Jt}=\mathrm{e}^{\lambda t} \sum_{k=0}^{p-1}\frac{t^kN^k}{k!}.
The forced system has the solution
x(t)=\mathrm{e}^{A(t-t_0)}x_0+ \int_{t_0}^t\mathrm{e}^{A(t-s)}b(s)\,ds.
If b is constant and A is invertible,
x_\star=-A^{-1}b,\qquad x(t)=x_\star+\mathrm{e}^{A(t-t_0)}(x_0-x_\star).
9 Higher-Order Scalar Linear ODEs
Consider
a_p(t)y^{(p)}+\cdots+a_0(t)y=g(t).
For constant coefficients, the characteristic polynomial is
p(\lambda)=a_p\lambda^p+\cdots+a_0.
A real root \lambda of multiplicity m gives
\mathrm{e}^{\lambda t},\quad t\mathrm{e}^{\lambda t},\quad\ldots, \quad t^{m-1}\mathrm{e}^{\lambda t}.
A conjugate pair \alpha\pm i\beta gives
\mathrm{e}^{\alpha t}\cos\beta t,\qquad \mathrm{e}^{\alpha t}\sin\beta t.
The superposition principle is
L[y]=g,\qquad y=y_h+y_p.
10 Autonomous Systems and Phase Space
For x'=f(x), an equilibrium x_\star satisfies
f(x_\star)=0.
A set S is positively invariant if x_0\in S implies \varphi_t(x_0)\in S for every t\ge0. A first integral satisfies
\frac{d}{dt}I(x(t))=\nabla I(x)^Tf(x)=0.
11 Stability of Linear Systems
For x'=Ax,
\text{asymptotically/exponentially stable} \quad\Longleftrightarrow\quad \operatorname{Re}\lambda<0 \quad\text{for every }\lambda\in\operatorname{spec}(A).
If some eigenvalue has positive real part, the system is unstable. If all eigenvalues have nonpositive real part, stability requires the eigenvalues on the imaginary axis to be semisimple.
12 Gradient, Conservative, and Hamiltonian Systems
For a gradient flow,
x'=-\nabla V(x),\qquad \frac{dV}{dt}=-\|\nabla V(x)\|^2\le0.
A Hamiltonian system has the form
z'=J\nabla H(z),\qquad J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}.
If H is autonomous, then dH/dt=0. The flow is symplectic:
(D\varphi_t)^TJ(D\varphi_t)=J.
13 Boundary-Value Problems
A boundary-value problem (BVP) is
x'=f(t,x),\qquad B(x(a),x(b))=0.
The linear case is
x'=A(t)x+b(t),\qquad B_ax(a)+B_bx(b)=c.
Shooting parameterizes the missing initial data and solves a residual equation F(s)=0. Multiple shooting uses several IVPs linked by continuity conditions. Collocation enforces the ODE at selected points of a finite-dimensional approximation.
14 Final Concept Map
\boxed{x'=f(t,x)\Longrightarrow \{\text{existence, uniqueness, flow, sensitivity, stability, numerics}\}.}
\boxed{x'=A(t)x+b(t)\Longrightarrow \{\Phi(t,s),\text{ Liouville, variation of constants}\}.}
\boxed{x'=Ax\Longrightarrow \{\mathrm{e}^{At},\operatorname{spec}(A),\text{ Jordan form, linear stability}\}.}
\boxed{x'=f(x),\ f(x_\star)=0\Longrightarrow \{\text{equilibrium, invariants, autonomous dynamics}\}.}