Mathematics \ Differential Equations \ First Order Equations

First order differential equations form a fundamental category within the study of differential equations and are essential in numerous applications across various fields such as physics, biology, and economics. A first order differential equation is an equation that involves the first derivative of an unknown function and can be generally represented as:

\[ \frac{dy}{dx} = f(x, y) \]

Here, \( y = y(x) \) is the unknown function we seek to determine, and \( f(x, y) \) is a given function of the independent variable \( x \) and the dependent variable \( y \).

Types of First Order Differential Equations

1. Separable Equations: These are equations that can be written in the form:

\[
\frac{dy}{dx} = g(x)h(y)
\]

This form allows the variables to be separated and integrated individually. Specifically, we can rewrite the equation as:

\[
\frac{1}{h(y)} \, dy = g(x) \, dx
\]

Integration of both sides then yields the solution:

\[
\int \frac{1}{h(y)} \, dy = \int g(x) \, dx + C
\]

where \( C \) is the constant of integration.

2. Linear First Order Equations: These have the general form:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

Such equations can be solved using an integrating factor \( \mu(x) \) defined as:

\[
\mu(x) = e^{\int P(x) \, dx}
\]

Multiplying both sides of the differential equation by the integrating factor transforms it into a form that is easily solvable as a product rule:

\[
\mu(x) \frac{dy}{dx} + \mu(x) P(x) y = \mu(x) Q(x)
\]

This simplifies to:

\[
\frac{d}{dx} [ \mu(x) y ] = \mu(x) Q(x)
\]

Integration of both sides provides the solution:

\[
\mu(x) y = \int \mu(x) Q(x) \, dx + C
\]

Thus,

\[
y = \frac{1}{\mu(x)} \left( \int \mu(x) Q(x) \, dx + C \right)
\]

3. Exact Equations: An exact differential equation is one that can be written in the form:

\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]

The equation is exact if there exists a function \( \Psi(x, y) \) such that:

\[
\frac{\partial \Psi}{\partial x} = M \quad \text{and} \quad \frac{\partial \Psi}{\partial y} = N
\]

That implies the differential \( d\Psi = M \, dx + N \, dy \). The solution can be found by integrating \( M \) with respect to \( x \) and \( N \) with respect to \( y \), and then determining the function \( \Psi(x, y) = C \).

Examples and Applications

First order differential equations serve as models for a wide array of scenarios. For instance:
- In population dynamics, the rate of change of a population \( P \) with respect to time \( t \) might be proportional to the current population, modeled as \( \frac{dP}{dt} = kP \), where \( k \) is a constant.
- In thermal physics, the rate of heat loss of a body can be represented by Newton’s Law of Cooling, which is a first order linear differential equation given by \( \frac{dT}{dt} = -k(T - T_{\text{env}}) \), where \( T \) is the temperature of the body, \( T_{\text{env}} \) is the ambient temperature, and \( k \) is a constant.

In conclusion, first order differential equations are foundational within the broader field of differential equations, offering analytical techniques and insights essential for modeling and solving problems in diverse scientific disciplines.