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DIFFERENTIAL EQUATIONS CHEAT SHEET

jeffdecola.com MIT License

The study of continuous change.

Table of Contents

Documentation and Reference

OVERVIEW

  • Calculus is a broad field of mathematics that includes differentiation (finding derivatives) and integration (finding integrals). It focuses on rates of change and accumulation.

  • Differential Equations (DiffEQ) are a specific branch of mathematics that deals with equations involving derivatives. A differential equation expresses a relationship between a function and its derivatives. Differential equations are used to model real-world phenomena involving rates of change and accumulation.

It's important to understand the role of $f(x)$ in both calculus and differential equations.

CONTEXT WHAT f(x) REPRESENTS WHAT WE WANT TO FIND
CALCULUS (Derivatives) $f(x)$ is the original function The derivative $f'(x)=\frac{dy}{dx}$
CALCULUS (Integrals) $f(x)$ is function to integrate Function $F(x)=\int f(x)dx + C$
DIFFERENTIAL EQUATIONS $f(x)$ is the derivative of $y(x)$ Solve for $y(x)$ by integrating

It's also important to understand the notation,

  • $dx$ means a small change in x
  • $dy$ means a small change in y
  • $\frac{dy}{dx}$ means the rate of change of y with respect to x

HOW TO SOLVE A DIFFERENTIAL EQUATION

The goal of diffEQ is to find the function y(x) that satisfies the
differential equation.

Given this first-order ordinary differential equation let's walk through the steps to solve it.

$$ \frac{dy}{dx} = 2x $$

Separate the variables,

$$ dy = 2x dx $$

Integrate both sides.

$$ \int dy = \int 2x dx $$

$$ y = x² + C $$

Solve for the constant C using initial conditions.

$$ y(0) = 0² + C = 0 $$

Hence

$$ C = 0 $$

Therefore, the solution to the differential equation is

$$ y = x² $$

UNDERSTANDING f(x)

Understanding the role of $f(x)$ is important because:

  • In calculus, you typically differentiate or integrate $f(x)$ to get new functions.
  • In differential equations, you start with $y'(x) = f(x) = \frac{dy}{dx}$ and integrate to recover $y(x)$.
  • The notation can be tricky, but knowing whether $f(x)$ is the function or its derivative helps avoid confusion.

f(x) in CALCULUS (DERIVATIVE)

Given a function, find the rate of change.

Original function,

$$ f(x) = y $$

Derivative of that function (find rate of change),

$$ f'(x) = \frac{dy}{dx} $$

As an example,

$$ f(x) = y = x² + 3x + 5 $$

The derivative of this function (rate of change) is,

$$ f'(x) = \frac{dy}{dx} = 2x + 3 $$

f(x) in CALCULUS (INTEGRAL)

Given the rate of change, find the function.

Original function,

$$ f(x) = \frac{dy}{dx} $$

Integral of that function (find function),

$$ F(x) = \int f(x) dx $$

As an example,

$$ f(x) = \frac{dy}{dx} = 2x + 3 $$

The integral of this function is (find function),

$$ F(x) = \int (2x +3 ) dx = x² + 3x + C $$

You may also see the integral written as,

$$ F(x) = y(x) $$

f(x) in DIFFERENTIAL EQUATIONS

In differential equations, f(x) is often used to represent the derivative of another function y(x). Like integrals, start with the rate of change and find the function.

$$ f(x) = \frac{dy}{dx} $$

Separate the variables

$$ f(x){dx} = {dy} $$

integrate both sides of the equation,

$$ \int f(x) dx = \int {dy} $$

$$ \int f(x) dx = y(x) $$

As an example,

$$ f(x) = \frac{dy}{dx} = 2x + 3 $$

or

$$ \frac{dy}{dx} = 2x + 3 $$

Separate the variables,

$$ dy = (2x + 3) dx $$

Integrate both sides,

$$ \int dy = \int (2x + 3) dx $$

$$ y(x) = x² + 3x + C $$

CLASSIFICATION OF DIFFERENTIAL EQUATIONS

Differential equations can be classified in many ways, we will classify them by type and order.

BY TYPE

  • Ordinary Differential Equations (ODEs) involve only one independent variable. For example,

$$ \frac{dy}{dx} = 2x $$

  • Partial Differential Equations (PDEs) involve more than one independent variable. For example,

$$ \frac{\partial u}{\partial t} = \frac{\partial² u}{\partial x²} $$

where $u$ is a function of $x$ and $t$.

BY ORDER

  • First-order differential equations involve only the first derivative. For example,

$$ y'(x) = \frac{dy}{dx} = 2x $$

  • Second-order differential equations involve the second derivative. For example,

$$ y''(x) = \frac{d²y}{dx²} = 2x $$

EXAMPLES

Since differential equations are used to model real-world phenomena, let's consider a simple example. Remember, the goal of differential equations is to find the function $y(x)$ that satisfies the equation.

POPULATION GROWTH

Consider a population of bacteria that grows at a rate proportional to the current population. The differential equation that models this growth is,

$$ P'(t)= \frac{dP}{dt} = kP $$

where $P$ is the population and $k$ is the growth rate constant. We want to find P(t), the population at a particular time.

To solve this differential equation, we integrate both sides,

$$ \int \frac{dP}{dt} dt = \int kP dt $$

$$ \int \frac{dP}{P} = \int k dt $$

$$ \ln P = kt + C $$

Solving for natural log we get

$$ P(t) = e^{kt + C} $$

$$ P(t) = P_0e^{kt} $$

where $P_0$ is the initial population at time $t=0$ and k is

$$ k = \frac{1}{t} ln \frac{P(t)}{P_0} $$

As an example, if we have an initial popular of 100 bacteria thats doubles every 3 hours,

$$ P_0 = 100 \quad and \quad P(3) = 2P_0 = 200 $$

the constant k would be

$$ k = \frac{1}{3} ln \frac{200}{100} = \frac{1}{3} ln 2 =0.231$$

Therefore, the population at time t would be,

$$ P(t) = 100e^{0.231t} $$