7.1 Modeling Situations with Differential Equations

• What is the first step when modeling a situation with differential equations?

  Identify quantities and their rates of change
• The independent variable in a differential equation changes at a constant rate.


  True
• The general form of a differential equation is $\frac{dy}{dt} =$$f(t, y)$, where `y` is the dependent variable
• The rate of change of an independent variable is constant.


  True
• What is the general form of a differential equation used to model relationships between quantities?

  \frac{dy}{dt} = f(t, y)</latex>
• The function $f(t, y)$ in the differential equation $\frac{dy}{dt} =$$f(t, y)$ describes how the rate of change of `y` depends on `t` and y
• The general form of a differential equation is \frac{dy}{dt}
• What does $f(t, y)$ describe in the general form of a differential equation?


  Rate of change of y
• The general form of a differential equation is \frac{dy}{dt}
• What does $f(t, y)$ describe in the general form of a differential equation?


  Rate of change of y
• To find a particular solution to a differential equation, you need to apply initial conditions
• What does $y_{0}$ represent in the initial condition $y(t_{0}) =$$y_{0}$?


  Value of y at $t_{0}$
• Given the differential equation \frac{dy}{dt} = 2t</latex> and the initial condition $y(0) =$$3$, the constant $C$ is equal to 3
• The independent variable in a differential equation typically represents the quantity that changes at a constant rate
• What variable is commonly used to denote the independent variable in differential equations?

  t
• What is the typical example of an independent variable in differential equations?

  Time
• In the differential equation $\frac{dy}{dt} =$$f(t, y)$, the term $\frac{dy}{dt}$ represents the rate of change of `y` with respect to t
• In the differential equation $\frac{dy}{dt} =$$f(t, y)$, what does $\frac{dy}{dt}$ represent?


  Rate of change
• Steps to translate relationships into a differential equation

  1️⃣ Identify the independent and dependent variables

  2️⃣ Determine the relationship between the variables

  3️⃣ Express the relationship using the general form of the differential equation
• In the general form of a differential equation, $\frac{dy}{dt}$ represents the rate of change of the dependent variable $y$ with respect to the independent variable $t$.


  True
• Steps to translate relationships into a differential equation

  1️⃣ Identify the independent and dependent variables

  2️⃣ Determine the relationship between the variables

  3️⃣ Express the relationship using the general form
• In the general form of a differential equation, $\frac{dy}{dt}$ represents the rate of change of the dependent variable $y$ with respect to the independent variable $t$.


  True
• Steps to translate relationships into a differential equation

  1️⃣ Identify the independent and dependent variables

  2️⃣ Determine the relationship between the variables

  3️⃣ Express the relationship using the general form
• Initial conditions are typically expressed as $y(t_{0}) =$$y_{0}$, where $t_{0}$ is the value of the independent variable.


  True
• Steps to apply initial conditions to find a particular solution

  1️⃣ Find the general solution

  2️⃣ Apply the initial condition to determine the constant

  3️⃣ Write the particular solution
• The general solution of the differential equation $\frac{dy}{dt} =$$2t$ is $y =$$t^{2} +$$C$.


  True
• Steps to apply initial conditions to find a particular solution

  1️⃣ Find the general solution

  2️⃣ Apply the initial condition to determine the constant

  3️⃣ Write the particular solution
• The initial condition $y(0) =$$3$ means that $y$ equals 3 when $t$ is 0.


  True
• Match the quantity type with its notation:

  Independent Variable ↔️ $t$

  Dependent Variable ↔️ $y$
• Understanding the relationships between variables is the first step in setting up differential equations.


  True
• The general form of a differential equation is \frac{dy}{dt}
• What should you determine when identifying relationships between quantities in a differential equation?

  How y changes with t
• When identifying relationships, you should determine how the dependent variable changes with the independent variable.
• In a differential equation, $\frac{dy}{dt}$ represents the rate of change of the dependent variable $y$ with respect to the independent variable $t$.


  True
• Match the relationship type with its mathematical expression:

  Linear ↔️ $\frac{dy}{dt} =$$k$

  Exponential ↔️ $\frac{dy}{dt} =$$ky$

  Inverse ↔️ $\frac{dy}{dt} =$$\frac{k}{y}$
• The mathematical expression $\frac{dy}{dt} =$$ky$ represents exponential growth in a differential equation.


  True
• Initial conditions are used to find a particular solution to a differential equation.
• What type of relationship does the general form of a differential equation express?

  Independent and dependent
• What is the mathematical expression for a linear relationship in a differential equation?

  $\frac{dy}{dt} =$$k$
• What is the purpose of setting up differential equations in this context?

  To model situations