7.1 Modeling Situations with Differential Equations

• Ordinary Differential Equations (ODEs) have one independent variable
• Partial Differential Equations (PDEs) use partial derivatives.
• What is the primary purpose of the separation of variables method?

  To isolate dependent and independent variables
• In the separation of variables method, the equation is rewritten in the form g(y) dy = f(x) dx
• Integrating both sides of $\frac{dy}{y} =$$x dx$ results in \ln|y| = \frac{1}{2}x^{2} + C</latex>.
• Steps for solving a differential equation using separation of variables

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for y
• What is the purpose of applying initial conditions to a differential equation?

  To find a particular solution
• To find the value of the constant $C$, initial conditions such as $y(0) =$$2$ are applied
• What must be applied to the general solution of a differential equation to find a particular solution?

  Initial conditions
• Steps to find a particular solution of a differential equation

  1️⃣ Find the general solution

  2️⃣ Apply initial conditions

  3️⃣ Write the particular solution
• What is the general solution of the differential equation \frac{dy}{dx} = 3x^{2}</latex>?

  $y(x) =$$x^{3} +$$C$
• Differential equations can be used to model population growth and decay.
• What is a differential equation?

  An equation relating a function to its derivatives
• Differential equations help model real-world phenomena such as population growth, radioactive decay, and Newton's Law of Cooling
• The differential equation $dP / dt =$$kP$ models the rate of growth of a population.
• What does a differential equation involve?

  A function and its derivatives
• Newton's Law of Cooling describes the rate of temperature change based on the temperature difference between an object and its surroundings
• Order the steps to solve a basic differential equation using separation of variables.

  1️⃣ Separate the variables

  2️⃣ Integrate both sides

  3️⃣ Apply initial conditions if given
• What is the independent variable in a differential equation?

  The variable being differentiated
• The dependent variable in a differential equation is the variable being studied.
• The derivative in a differential equation represents the rate of change of the dependent variable with respect to the independent variable
• In the differential equation $dP / dt =$$kP$, what is the dependent variable?


  $P$
• In the differential equation $dP / dt =$$kP$, what is the independent variable?


  $t$
• What is a derivative in the context of differential equations?

  Rate of change
• In the differential equation \frac{dP}{dt} = kP</latex>, the dependent variable is P
• What are the components of a differential equation?

  Function and derivatives
• The differential equation for population growth is \frac{dP}{dt} = kP
• Newton's Law of Cooling involves a negative sign in its differential equation because temperature decreases over time.
• What does a differential equation relate?

  A function to its derivatives
• Match the real-world phenomenon with its corresponding description:

  Population Growth ↔️ Rate of population change depends on current population

  Newton's Law of Cooling ↔️ Rate of temperature change depends on temperature difference

  Radioactive Decay ↔️ Rate of decay is proportional to remaining material
• What does the constant $k$ represent in the population growth differential equation $\frac{dP}{dt} =$$kP$?


  Growth rate
• In Newton's Law of Cooling, the term T_{s}</latex> represents the surrounding temperature.
• What is the order of a differential equation?

  Highest order derivative
• Match the differential equation with its order and degree:

  $y' +$$2y =$$x$ ↔️ Order: 1, Degree: 1

  $y'' - 4y' +$$3y =$$0$ ↔️ Order: 2, Degree: 1

  $(y'')^{2} +$$(y')^{3} =$$x^{2}$ ↔️ Order: 2, Degree: 2

  $\frac{d^{3}y}{dx^{3}} +$$5\frac{d^{2}y}{dx^{2}} - 2\frac{dy}{dx} =$$0$ ↔️ Order: 3, Degree: 1
• What is the order of a differential equation?

  Highest order derivative
• The degree of a differential equation is the power to which the highest order derivative is raised after the equation has been rationalized
• Match the differential equation with its order and degree:

  y' + 2y = x</latex> ↔️ Order: 1, Degree: 1

  $y'' - 4y' +$$3y =$$0$ ↔️ Order: 2, Degree: 1

  $(y'')^{2} +$$(y')^{3} =$$x^{2}$ ↔️ Order: 2, Degree: 2

  $\frac{d^{3}y}{dx^{3}} +$$5\frac{d^{2}y}{dx^{2}} - 2\frac{dy}{dx} =$$0$ ↔️ Order: 3, Degree: 1
• Ordinary Differential Equations (ODEs) involve functions of multiple independent variables.

  False
• Ordinary Differential Equations (ODEs) involve functions of a single independent variable
• Partial Differential Equations (PDEs) use ordinary derivatives.

  False