7.8 Exponential Models with Differential Equations

• What is the general form of a differential equation representing exponential growth or decay?

  $\frac{dy}{dt} =$$ky$
• In the differential equation $\frac{dy}{dt} =$$ky$, $k$ is called the constant of proportionality
• The solution to the differential equation $\frac{dy}{dt} =$$ky$ is $y(t) =$$y_{0}e^{kt}$.
• In the solution $y(t) =$$y_{0}e^{kt}$, $y_{0}$ represents the initial value
• If a bacterial population doubles every 2 hours, what is the value of the constant k</latex> in the exponential model?

  $\frac{\ln 2}{2}$
• In exponential growth, the constant of proportionality $k$ is positive.
• In exponential decay, the constant of proportionality $k$ is negative
• What is the significance of a positive constant of proportionality $k$ in an exponential model?


  Describes exponential growth
• If the constant $k$ in the differential equation $\frac{dy}{dt} =$$ky$ is greater than zero, the model describes exponential growth.
• Steps to solve an exponential differential equation of the form \frac{dy}{dt} = ky</latex>

  1️⃣ Separate the variables: $\frac{dy}{y} =$$k dt$.

  2️⃣ Integrate both sides: $\ln|y| =$$kt +$$C$.

  3️⃣ Solve for $y$: $y(t) =$$y_{0}e^{kt}$.
• Consider a bacterial population growing according to $\frac{dP}{dt} =$$0.04P$ with an initial population of $500$. What is the solution for $P(t)$?


  $P(t) =$$500e^{0.04t}$
• Steps to solve exponential differential equations of the form $\frac{dy}{dt} =$$ky$


  1️⃣ Separation of Variables: $\frac{dy}{y} =$$k dt$

  2️⃣ Integration: $\int \frac{dy}{y} =$$\int k dt$, resulting in $\ln|y| =$$kt +$$C$

  3️⃣ Solve for y: $y(t) =$$y_{0}e^{kt}$, where $y_{0} =$$e^{C}$
• What is the general solution to $\int \frac{dy}{y} =$$\int k dt$?


  $\ln|y| =$$kt +$$C$
• The initial value y_{0}</latex> in the solution $y(t) =$$y_{0}e^{kt}$ is equal to $e^{C}$
• Consider a bacterial population that grows according to $\frac{dP}{dt} =$$0.04P$ with an initial population of $500$ bacteria. The value of $k$ in this case is 0.04
• What is the separated form of the equation \frac{dP}{dt} = 0.04P</latex>?

  $\frac{dP}{P} =$$0.04 dt$
• Integrating $\frac{dP}{P} =$$0.04 dt$ results in $\ln|P| =$$0.04t +$$C$
• What is the population model for a bacterial population that grows according to $\frac{dP}{dt} =$$0.04P$ with an initial population of $500$ bacteria?


  $P(t) =$$500e^{0.04t}$
• Exponential models describe phenomena where the rate of change is proportional to the quantity
• What is the differential equation that represents an exponential model?

  \frac{dy}{dt} = ky</latex>
• In a bacterial population that doubles every 2 hours, the exponential model includes the constant $\ln 2$
• What is the key parameter in the differential equation $\frac{dy}{dt} =$$ky$?


  $k$
• Match the value of k</latex> with the type of exponential model:

  $k > 0$ ↔️ Exponential Growth

  $k < 0$ ↔️ Exponential Decay
• What are the three phases in solving an exponential differential equation?

  Separation, Integration, Solve
• When integrating $\frac{dy}{y} =$$k dt$, the result is \ln|y| = kt + C
• Exponential models can describe population growth under ideal conditions.
• What is the formula for exponential decay when $k > 0$?


  $y(t) =$$y_{0} e^{ - kt}$
• Match the real-world scenario with its exponential model:

  Bacteria doubling every hour ↔️ $P(t) =$$100 \cdot 2^{t}$

  Radioactive decay at 8.7% per day ↔️ $R(t) =$$500 e^{ - 0.087t}$

  Compound interest at 5% annually ↔️ $A(t) =$$1000 e^{t \ln 1.05}$