Calculus Volume 2

# Key Concepts

Calculus Volume 2Key Concepts

### 4.1Basics of Differential Equations

• A differential equation is an equation involving a function $y=f(x)y=f(x)$ and one or more of its derivatives. A solution is a function $y=f(x)y=f(x)$ that satisfies the differential equation when $ff$ 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.

### 4.2Direction Fields and Numerical Methods

• 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.

### 4.3Separable Equations

• A separable differential equation is any equation that can be written in the form $y′=f(x)g(y).y′=f(x)g(y).$
• The method of separation of variables is used to find the general solution to a separable differential equation.

### 4.4The Logistic Equation

• 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.

### 4.5First-order Linear Equations

• Any first-order linear differential equation can be written in the form $y′+p(x)y=q(x).y′+p(x)y=q(x).$
• 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.