Introduction to Differential Equations

Vocabulary and key mathematical forms related to the definition, classification, and solution methods of ordinary differential equations.

$n$-th Order Differential Equation (Implicit Form)

An equation relating an independent variable $x$, a function $y(x)$, and its derivatives up to the $n$-th order, expressed as $F(x, y, y', …, y^{(n)}) = 0$.

$n$-th Order Differential Equation (Explicit Form)

A differential equation where the highest order derivative is expressed as a function of lower order derivatives and the independent variable: $y^{(n)} = f(x, y, y', …, y^{(n-1)})$.

Solution of a Differential Equation

A function $u \in C^n(I)$ defined on an interval $I \subset R$ that satisfies the equation $y^{(n)} = f(x, y, y', \dots, y^{(n-1)})$ such that for all $x \in I$, $(x, u(x), u'(x), \dots, u^{(n-1)}(x)) \in \Omega$.

Separable Differential Equation

A first-order differential equation that can be written in the form $y' = f(t)g(y)$, which is solved by integrating both sides: $\int \frac{1}{g(y)}\,dy = \int f(t)\,dt + C$.

Exponential Growth Model

A fundamental differential equation represented by $\frac{dy}{dt} = ky(t)$.

Second-Order Differential Equation Example

An equation involving the second derivative of the unknown function, such as $\frac{d^2y}{dt^2} = -\sin(y(t))$.

Separable Example Equation

A specific case of a separable differential equation shown as $ty' = y$.

Geometric Family Differential Equation

The differential equation derived from the family of curves $(x - C)^2 + y = 1$, expressed in the transcript as $y'y'' + y' = 0$.