Basics of Differential Equations

• A differential equation is an equation involving a function [latex]y=f\left(x\right)[/latex] and one or more of its derivatives. A solution is a function [latex]y=f\left(x\right)[/latex] that satisfies the differential equation when [latex]f[/latex] and its derivatives are substituted into the equation.
• The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.
• A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.

Direction Fields and Euler’s Method

• A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.
• Euler’s Method is a numerical technique that can be used to approximate solutions to a differential equation.

Separation of Variables

• A separable differential equation is any equation that can be written in the form [latex]y^{\prime} =f\left(x\right)g\left(y\right)[/latex].
• The method of separation of variables is used to find the general solution to a separable differential equation.

First-Order Linear Equations and Applications

• When studying population functions, different assumptions—such as exponential growth, logistic growth, or threshold population—lead to different rates of growth.
• The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.
• The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.
• Any first-order linear differential equation can be written in the form [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex].
• We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.
• Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.

Key Equations

• Euler’s Method
  
  [latex]\begin{array}{c}{x}_{n}={x}_{0}+nh\hfill \\ {y}_{n}={y}_{n - 1}+hf\left({x}_{n - 1},{y}_{n - 1}\right),\text{where}h\text{is the step size}\hfill \end{array}[/latex]
• Separable differential equation
  
  [latex]{y}^{\prime }=f\left(x\right)g\left(y\right)[/latex]
• Solution concentration
  
  [latex]\frac{du}{dt}=\text{INFLOW RATE}-\text{OUTFLOW RATE}[/latex]
• Newton’s law of cooling
  
  [latex]\frac{dT}{dt}=k\left(T-{T}_{s}\right)[/latex]
• Logistic differential equation and initial-value problem
  
  [latex]\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),P\left(0\right)={P}_{0}[/latex]
• Solution to the logistic differential equation/initial-value problem
  
  [latex]P\left(t\right)=\frac{{P}_{0}K{e}^{rt}}{\left(K-{P}_{0}\right)+{P}_{0}{e}^{rt}}[/latex]
• Threshold population model
  
  [latex]\frac{dP}{dt}=\text{-}rP\left(1-\frac{P}{K}\right)\left(1-\frac{P}{T}\right)[/latex]
• standard form
  
  [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex]
• integrating factor
  
  [latex]\mu \left(x\right)={e}^{\displaystyle\int p\left(x\right)dx}[/latex]

Glossary

asymptotically semi-stable solution

[latex]y=k[/latex] if it is neither asymptotically stable nor asymptotically unstable

asymptotically stable solution

[latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] approaches [latex]k[/latex] as [latex]x[/latex] approaches infinity

asymptotically unstable solution

[latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] never approaches [latex]k[/latex] as [latex]x[/latex] approaches infinity

autonomous differential equation

an equation in which the right-hand side is a function of [latex]y[/latex] alone

carrying capacity

the maximum population of an organism that the environment can sustain indefinitely

differential equation

an equation involving a function [latex]y=y\left(x\right)[/latex] and one or more of its derivatives

direction field (slope field)

a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point

equilibrium solution

any solution to the differential equation of the form [latex]y=c[/latex], where [latex]c[/latex] is a constant

Euler’s Method

a numerical technique used to approximate solutions to an initial-value problem

general solution (or family of solutions)
the entire set of solutions to a given differential equation

growth rate

the constant [latex]r>0[/latex] in the exponential growth function [latex]P\left(t\right)={P}_{0}{e}^{rt}[/latex]

initial population

the population at time [latex]t=0[/latex]

initial value(s)

a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable

initial velocity

the velocity at time [latex]t=0[/latex]

initial-value problem

a differential equation together with an initial value or values

integrating factor

any function [latex]f\left(x\right)[/latex] that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions

linear

description of a first-order differential equation that can be written in the form [latex]a\left(x\right){y}^{\prime }+b\left(x\right)y=c\left(x\right)[/latex]

logistic differential equation

a differential equation that incorporates the carrying capacity [latex]K[/latex] and growth rate [latex]r[/latex] into a population model

order of a differential equation

the highest order of any derivative of the unknown function that appears in the equation

particular solution

member of a family of solutions to a differential equation that satisfies a particular initial condition

phase line

a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions

separable differential equation

any equation that can be written in the form [latex]y^{\prime} =f\left(x\right)g\left(y\right)[/latex]

separation of variables

a method used to solve a separable differential equation

solution curve

a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field

solution to a differential equation

a function [latex]y=f\left(x\right)[/latex] that satisfies a given differential equation

standard form

the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex]

step size

the increment [latex]h[/latex] that is added to the [latex]x[/latex] value at each step in Euler’s Method

threshold population

the minimum population that is necessary for a species to survive