7.1 Modeling Situations with Differential Equations

• What is a differential equation?

  Relates a function and derivatives
• Match the type of differential equation with its definition:

  Ordinary Differential Equation (ODE) ↔️ Equation with one independent variable

  Partial Differential Equation (PDE) ↔️ Equation with multiple independent variables
• Give an example of a Partial Differential Equation (PDE).

  $\frac{\partial u}{\partial t} =$$\frac{\partial^{2} u}{\partial x^{2}}$
• In a differential equation, the dependent variable is the function we aim to find
• In the differential equation \frac{dy}{dt} = k \cdot y</latex>, what is the dependent variable?

  y
• Steps in translating real-world scenarios into differential equations:

  1️⃣ Identify variables

  2️⃣ Define rate of change

  3️⃣ Formulate equation
• What is the derivative of $x^{n}$?


  $nx^{n - 1}$
• Steps involved in mathematical modeling using differential equations

  1️⃣ Identify Variables

  2️⃣ Define Rate of Change

  3️⃣ Formulate Equation
• What does the rate of change describe in a differential equation?

  How variables affect each other
• Match the function type with its derivative:

  Constant ↔️ 0

  Power ↔️ $nx^{n - 1}$

  Exponential ↔️ $e^{x}$

  Logarithmic ↔️ $\frac{1}{x}$
• What is the derivative of $x^{n}$?


  $nx^{n - 1}$
• What is the derivative of $\ln(x)$?


  $\frac{1}{x}$
• A differential equation relates a function and its derivatives


  True
• What is an example of an ordinary differential equation (ODE)?

  \frac{dy}{dx} = 2x</latex>
• What is the dependent variable in a differential equation?

  Function being sought
• In the equation $\frac{dP}{dt} =$$kP$, the dependent variable is P
• What is the differential equation for a population growth rate proportional to the current population?

  $\frac{dP}{dt} =$$kP$
• What is the derivative of a constant, c, with respect to x?

  0
• What is the derivative of $e^{x}$ with respect to x?


  $e^{x}$
• Steps for solving the differential equation $\frac{dy}{dx} =$$\frac{x}{y}$ using separation of variables.


  1️⃣ $\frac{dy}{dx} =$$\frac{x}{y}$

  2️⃣ $y \, dy =$$x \, dx$

  3️⃣ $\int y \, dy =$$\int x \, dx$
• If P(0) = P_{0}</latex> in the population growth model, what is $P(t)$?


  $P_{0} e^{kt}$
• What is the sign of k</latex> for exponential growth?

  Positive
• The solution for exponential decay is P(t) = P_{0} e^{-kt}</latex>, where the exponent is negative
• Steps for modeling exponential growth with $\frac{dP}{dt} =$$kP$


  1️⃣ Identify the dependent and independent variables

  2️⃣ Determine the growth rate k

  3️⃣ Find the initial condition P_0

  4️⃣ Solve the differential equation

  5️⃣ Interpret the solution
• In exponential decay, the value of k is negative
• Match the sign of k with its effect on the quantity:


  k is positive ↔️ Growth

  k is negative ↔️ Decay
• The solution to the differential equation $\frac{dP}{dt} =$$0.05P$ is P(t) = 100 e^{0.05t}
• What does the dependent variable P represent in the differential equation $\frac{dP}{dt} =$$kP$?


  Quantity of interest
• For exponential growth, the value of k is positive.


  True
• In the exponential growth solution $P(t) =$$P_{0} e^{kt}$, the term P_0 represents the initial value
• What is the solution for exponential decay given the differential equation $\frac{dP}{dt} =$$kP$?


  $P(t) =$$P_{0} e^{ - kt}$
• What is the solution to the differential equation $\frac{dP}{dt} =$$0.05P$ with initial population $P_{0} =$$100$?


  $P(t) =$$100 e^{0.05t}$
• What is an example of an ordinary differential equation (ODE)?

  $\frac{dy}{dx} =$$2x$
• ODEs deal with relationships in one dimension, while PDEs extend to multiple dimensions.


  True
• In the differential equation \frac{dP}{dt} = kP</latex>, time $t$ is the independent variable.
• What is the derivative of a constant function $\frac{d}{dx}(c)$?


  0
• What is the primary method for solving the differential equation $\frac{dy}{dx} =$$\frac{x}{y}$?


  Separation of variables
• If $P(0) =$$P_{0}$, the solution to $\frac{dP}{dt} =$$kP$ is $P(t) =$$P_{0} e^{kt}$, which shows exponential growth.
• What differential equation is commonly used to model exponential growth and decay?

  $\frac{dP}{dt} =$$kP$
• Match the exponential growth and decay solutions with their respective growth rates:

  Exponential growth ↔️ P(t) = P_{0} e^{kt}</latex>

  Exponential decay ↔️ $P(t) =$$P_{0} e^{ - kt}$