7.1 Modeling Situations with Differential Equations

• Solving differential equations often requires techniques like integration
• Identifying the type of differential equation is important as it determines the appropriate solution techniques
• An Ordinary Differential Equation (ODE) involves a function of a single independent variable.


  True
• The equation d²x/dt² = -gx is a second-order ODE describing motion under constant gravity.


  True
• Population growth over time can be modeled using a first-order ODE such as dy/dt = ky
• Heat distribution in a 2D object over time can be modeled using a PDE
• What is a differential equation?

  Equation relating function to derivatives
• What is the defining characteristic of an ODE?

  Single independent variable
• What type of ODE is used to model exponential population growth?

  First-order ODE
• What is the first step in translating real-life scenarios into differential equation models?

  Identify key quantities
• Dependent variables in a differential equation model are quantities being modeled that change with respect to the independent variable(s).
• Separation of variables is used for first-order ODEs in the form \( dy/dx = f(x)g(y) \), where the variables can be separated to solve for \( y \).
• Numerical verification uses a numerical solver to approximate the solution and compares it to the given solution.
• A differential equation is an equation that relates a function to its derivatives
• A linear ODE has derivatives that appear nonlinearly.

  False
• Match the differential equation type with its independent variable:

  ODE ↔️ Single independent variable

  PDE ↔️ Multiple independent variables
• The equation dy/dt = ky represents a first-order ODE describing population growth
• Steps to determine the type of a differential equation

  1️⃣ Examine the relationships between quantities

  2️⃣ Classify as ODE or PDE

  3️⃣ Identify the order

  4️⃣ Determine the linearity
• What is the first step in translating real-life scenarios into differential equation models?

  Identify key quantities
• Translating real-life scenarios into differential equation models involves expressing relationships between quantities and their derivatives.


  True
• PDEs involve functions of two or more independent variables.


  True
• A second-order ODE contains both the first and second derivatives of the dependent variable.


  True
• The acceleration of an object under gravity can be modeled using a second-order ODE.


  True
• Match the scenario with the correct type of differential equation:

  Population growth over time ↔️ First-order ODE

  Motion of an object under constant force ↔️ Second-order ODE

  Heat distribution in a 2D object over time ↔️ PDE
• Steps involved in solving a differential equation using separation of variables

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for \( y \)
• What is the purpose of analytical verification in differential equations?

  To confirm equality
• Partial Differential Equations (PDEs) involve functions of multiple independent variables.


  True
• Partial Differential Equations (PDEs) involve functions of multiple independent variables.


  True
• Steps for translating a real-life scenario into a differential equation model:

  1️⃣ Identify key quantities and their relationships

  2️⃣ Determine the type of differential equation
• Translating real-life scenarios into differential equation models requires identifying the key quantities and their relationships
• The motion of an object under constant force can be described by a second-order ODE


  True
• Match the scenario with the type of differential equation:

  Population growth ↔️ First-order ODE

  Motion under constant force ↔️ Second-order ODE

  Heat distribution in 2D ↔️ PDE
• An example of an ODE is dy/dt = ky
• A first-order ODE contains only the first derivative
• The exponential growth model is represented by the first-order ODE dy/dt = ky
• Steps involved in translating real-life scenarios into differential equation models

  1️⃣ Identify the key quantities and their relationships

  2️⃣ Determine the type of differential equation

  3️⃣ Translate the key features into the differential equation
• Parameters in a differential equation model are constant coefficients that influence the dynamics of the model.


  True
• Match the method for checking solutions with its description:

  Analytical Verification ↔️ Substitute solution into the differential equation

  Numerical Verification ↔️ Use a numerical solver to approximate