1: Definitions and classification of ODEs

Section 1.1: Definitions and Classification of ODEs

An Ordinary Differential Equation (ODE) is an equation involving an unknown function of a single variable, one or more of its derivatives, and possibly the independent variable itself. The general form is $F(x, y, y', y'', \dots, y^{(n)}) = 0$, where $y$ is the unknown function of $x$, and $y^{(k)}$ denotes its $k$-th derivative.

The order of an ODE is the highest derivative present. For example:

• $\frac{dy}{dx} = 2x$ is a first-order ODE (highest derivative is $y'$).
• $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - y = 0$ is a second-order ODE (highest derivative is $y''$).

A solution to an ODE on an interval $I$ is a function $\phi(x)$ that satisfies the equation identically for all $x$ in $I$. Solutions can be expressed:

• Explicitly: $y = \phi(x)$ (e.g., $y = x^2 + C$ for $y' = 2x$).
• Implicitly: As an equation $G(x, y) = 0$ (e.g., $x^2 + y^2 - C = 0$ for $y' = -x/y$).

The general solution of an $n$-th order ODE typically contains $n$ arbitrary constants (e.g., $y = C_1e^x + C_2e^{-x}$ for $y'' - y = 0$). A particular solution is obtained by assigning specific values to these constants, often determined by initial conditions (e.g., setting $y(0)=1$ and $y'(0)=0$ gives $y = \cosh(x)$ for the previous equation). A singular solution is a solution not obtainable from the general solution by specifying constants; it often lies on the envelope of the family of general solution curves (e.g., $y = 0$ for $(y')^2 = 4y$).

ODEs are classified by linearity:

• A linear ODE can be written in the standard form:
  $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \dots + a_1(x)y' + a_0(x)y = g(x)$
  The key feature is that the unknown function $y$ and its derivatives appear only to the first power and are not multiplied together. The coefficients $a_k(x)$ and the nonhomogeneous term $g(x)$ are functions of $x$ only. If $g(x) = 0$, the equation is homogeneous linear; otherwise, it is nonhomogeneous linear.
• A nonlinear ODE is any equation that cannot be written in the linear form above. This occurs if $y$ or one of its derivatives appears:
  • Raised to a power other than 1 (e.g., $(y')^2 + y = 0$).
  • Inside a nonlinear function (e.g., $\sin(y') + y = 0$).
  • Multiplied by another derivative or function of $y$ (e.g., $yy'' + y' = 0$).