7.1 Modeling Situations with Differential Equations

• What is a differential equation?

  Relates function to its derivative(s)
• Newton's law of cooling explains how an object cools down to match the ambient temperature
• What is the general form of a first-order linear ODE?

  dy/dx + p(x)y = q(x)
• An autonomous differential equation has the form dy/dx = f(y), where f(y) depends on x.

  False
• Separable differential equations are often used to model exponential growth or decay
• A differential equation relates a function to its derivative(s).

  True
• A first-order linear ODE has the form dy/dx + p(x)y = q(x), where p(x) and q(x) are functions of x
• Match the type of differential equation with its form:

  First-Order Linear ODE ↔️ dy/dx + p(x)y = q(x)

  Separable Differential Equation ↔️ dy/dx = f(x)g(y)

  Autonomous Differential Equation ↔️ dy/dx = f(y)
• What is the general form of an autonomous differential equation?

  $\frac{dy}{dx} =$$f(y)$
• First-order linear ODEs have the form \(\frac{dy}{dx} + p(x)y = q(x)\).


  True
• Separable differential equations are primarily used to model exponential growth and decay.

  True
• Translating real-world scenarios into differential equations begins with identifying the key variables.
• The autonomous differential equation for population growth can be written as \frac{dP}{dt} = f(P)</latex>.


  True
• Why are initial conditions necessary for solving differential equations?

  Finding particular solutions
• A differential equation relates a function to its derivative(s).


  True
• A differential equation relates a function to its derivative
• The equation d²y/dx² = -y relates a function to its second derivative.


  True
• What is an example of a PDE?

  ∂u/∂x + ∂u/∂y = 0
• Differential equations are used in applications such as modeling population growth
• Separable differential equations allow variables to be separated for integration
• Steps to translate real-world scenarios into differential equations

  1️⃣ Identify key variables

  2️⃣ Analyze the rate of change

  3️⃣ Choose the appropriate type

  4️⃣ Formulate the differential equation
• What are initial conditions used for in differential equations?

  Finding particular solutions
• If a population starts at 500, the initial condition is \(P(0) = 500\).


  True
• What is the integrating factor for a first-order linear ODE?

  $\mu(x) =$ e^{\int p(x) dx}\)
• Numerical methods are employed when exact solutions are difficult to find
• Numerical methods approximate solutions at discrete points in time


  True
• What is the integrating factor for the equation $\frac{dy}{dx} +$$2xy =$$x$?


  $\mu(x) =$$e^{x^{2}}$
• Separation of variables is suitable for equations that can be written as $f(x)dx =$$g(y)dy$


  True
• The integrating factor for $\frac{dy}{dx} +$$2xy =$$x$ is e^{x^2}
• Constants in differential equation solutions are determined by initial conditions

  True
• The differential equation dy/dx = 2x relates the function y to its first derivative
• What type of differential equation is d²y/dx² = -y?

  Second-order
• Match the type of differential equation with its description:

  Ordinary Differential Equation (ODE) ↔️ Contains only one independent variable

  Partial Differential Equation (PDE) ↔️ Contains multiple independent variables

  First-Order Differential Equation ↔️ Contains only the first derivative

  Second-Order Differential Equation ↔️ Contains up to the second derivative
• What is one application of differential equations in real-world scenarios?

  Population growth modeling
• A partial differential equation (PDE) contains only one independent variable.

  False
• Separable differential equations can be written as dy/dx = f(x)g(y), allowing variables to be separated
• In what type of real-world problem are first-order linear ODEs commonly used?

  Decay problems
• Match the type of differential equation with its example:

  Ordinary Differential Equation (ODE) ↔️ dy/dx = x + y

  Partial Differential Equation (PDE) ↔️ ∂u/∂x + ∂u/∂y = 0

  First-Order Differential Equation ↔️ dy/dx = xy

  Second-Order Differential Equation ↔️ d²y/dx² + 4y = 0
• Give an example of a first-order differential equation.

  dy/dx = xy
• What is the general form of a separable differential equation?

  dy/dx = f(x)g(y)