MATH2003 Lecture 2: First Order Homogeneous & Exact Equations

Comprehensive vocabulary flashcards covering First Order Homogeneous Equations, Differentials, Partial Derivatives, Exact Equations, and Integrating Factors from MATH2003 Lecture 2.

First Order Homogeneous Equation

A first order differential equation of the form $\frac{dy}{dx} = g\left(\frac{y}{x}\right)$, which can be transformed into a separable equation via a change of variables.

Homogeneous Equation Substitution

The variable substitution $u = \frac{y}{x}$ (or $y = ux$), which yields $\frac{dy}{dx} = u + x\frac{du}{dx}$ to reduce a homogeneous differential equation to separable form.

Differential of a Single-Variable Function ($dy$)

The dependent variable defined as the product of the derivative and $dx$ ($dy = f'(x)\,dx$), representing the vertical change along the tangent line that approximates the actual change $\Delta y$ for small values of $dx$.

Differential of a Two-Variable Function ($du$)

The total differential of a function $u(x, y)$, defined by the formula $du = \frac{\partial u}{\partial x}\,dx + \frac{\partial u}{\partial y}\,dy$.

Partial Derivative with Respect to $x$ ($z_x$ or $\frac{\partial z}{\partial x}$)

The rate of change or tangent slope of $z = f(x, y)$ in a plane parallel to the $xz$-plane, evaluated by holding $y$ constant and differentiating with respect to $x$.

Partial Derivative with Respect to $y$ ($z_y$ or $\frac{\partial z}{\partial y}$)

The rate of change or tangent slope of $z = f(x, y)$ in a plane parallel to the $yz$-plane, evaluated by holding $x$ constant and differentiating with respect to $y$.

Pure Second Order Partial Derivatives

Second-order partial derivatives taken twice with respect to the same variable, denoted by $z_{xx} = \frac{\partial^2 z}{\partial x^2}$ and $z_{yy} = \frac{\partial^2 z}{\partial y^2}$.

Mixed Second Order Partial Derivatives

Second-order partial derivatives taken with respect to one variable and then the other, denoted by $z_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right) = \frac{\partial^2 z}{\partial y\partial x}$ and $z_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right) = \frac{\partial^2 z}{\partial x\partial y}$.

Multivariable Chain Rule

The rule for differentiating a two-variable composite function $z = z(x, y)$ where $x = x(t)$ and $y = y(t)$, given by $\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$.

Exact Differential Equation

A first-order differential equation expressed in the form $M(x, y)\,dx + N(x, y)\,dy = 0$ that can be written as $du = 0$, where $du$ is the total differential of a function $u(x, y)$.

Test for Exactness

The necessary and sufficient condition stating that the differential equation $M(x, y)\,dx + N(x, y)\,dy = 0$ is exact if and only if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

General Solution of an Exact Equation

The implicit solution $u(x, y) = k$, where $u(x, y)$ satisfies both $u = \int M(x, y)\,dx$ (holding $y$ constant) and $u = \int N(x, y)\,dy$ (holding $x$ constant).

Integrating Factor

A non-zero function $F(x, y)$ that transforms a non-exact differential equation $M(x, y)\,dx + N(x, y)\,dy = 0$ into an exact equation when multiplied across all terms.