DIFFERENTIAL EQUATIONS CHEAT SHEET

The study of continuous change.

Table of Contents

• OVERVIEW
• HOW TO SOLVE A DIFFERENTIAL EQUATION
• UNDERSTANDING f(x)
  • f(x) in CALCULUS (DERIVATIVE)
  • f(x) in CALCULUS (INTEGRAL)
  • f(x) in DIFFERENTIAL EQUATIONS
• CLASSIFICATION OF DIFFERENTIAL EQUATIONS
  • BY TYPE
  • BY ORDER
• EXAMPLES
  • POPULATION GROWTH

Documentation and Reference

• calculus
• my-latex-renders
• latex math mode

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

$$dy=2xdx$$

Integrate both sides.

$$\int dy=\int 2xdx$$

$$y=x²+C$$

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 $y^{\prime }(x)=f(x)=\frac{dy}{dx}$ 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),

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

As an example,

$$f(x)=y=x²+3x+5$$

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

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

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,

$$\frac{\partial u}{\partial t}=\frac{\partial ²u}{\partial x²}$$

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

BY ORDER

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

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

• 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,

$$P^{\prime }(t)=\frac{dP}{dt}=kP$$

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,

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

$$\int \frac{dP}{P}=\int kdt$$

$$\ln P=kt+C$$

Solving for natural log we get

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

$$P(t)=P_0{e}^{kt}$$

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,

$$P_0=100andP(3)=2P_0=200$$

the constant k would be

$$k=\frac{1}{3}ln\frac{200}{100}=\frac{1}{3}ln2=0.231$$

Therefore, the population at time t would be,

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