Differential Equations

Reviewing some basic ideas in differential equations.

1 Background

In this handout, we consider several simple differential equations that characterize growth. We then consider the solution of each differential equation in terms of \(y\) as a function of \(x\).

For logistic growth, I also consider a \(dp/dx\) formulation.

2 Constant (Linear) Growth

There is a constant rate of growth, expressed by \(k\).

Code

x <- seq(1,10)

y1 <- as.numeric(x)

mydata1 <- data.frame(x, y1)

plinear <- ggplot(data = mydata1,
                  aes(x = x,
                      y = y1,
                      group = 1)) +
  geom_line(linewidth = 2, color = "grey") +
  geom_point(aes(frame = x), color = "deepskyblue", size = 3) +
  labs(title = "Constant Growth") 

\[\frac{dy}{dx} = k\]

implies

\[y = kx + C\]

Code

plinearB <- plinear + 
  labs(subtitle = "y Grows Linearly As A Function Of x",
       y = "y")

ggplotly(plinearB)

3 Exponential Growth

Growth is a function of \(y\).

Code

x <- seq(1, 10, .5)

y2 <- exp(x)

mydata2 <- data.frame(x, y2)

pexponential <- ggplot(data = mydata2,
                       aes(x = x,
                           y = y2,
                           group = 1)) +
  geom_line(linewidth = 2, color = "grey") +
  geom_point(aes(frame = x), color = "deepskyblue", size = 3) +
  labs(title = "Exponential Growth") 

\[\frac{dy}{dx} = y\]

implies

\[y = e^x + C\]

Code

pexponentialB <- pexponential + 
  labs(subtitle = "y Grows Exponentially As A Function Of x",
       y = "y")

ggplotly(pexponentialB)

4 Logistic Growth

Growth is initially approximately exponential, but then tapers off as \(y\) approaches some limiting value.

Code

x <- seq(1, 20)

y3 <- exp(x - 10) / (1 + exp(x - 10))

mydata3 <- data.frame(x, y3)

plogistic <- ggplot(data = mydata3,
                       aes(x = x,
                           y = y3,
                           group = 1)) +
  geom_line(linewidth = 2, color = "grey") +
  geom_point(aes(frame = x), color = "deepskyblue", size = 3) +
  labs(title = "Logistic Growth") 

\(dy/dx\) Form

\[\frac{dy}{dx} = y \left(1 - y \right)\] implies

\[y = \frac{e^x}{1 + e^x} + C\]

Code

plogisticB <- plogistic + 
  labs(subtitle = "y Grows Exponentially Until y Approaches 1.0",
       y = "y")

ggplotly(plogisticB)

\(dp/dx\) Form

Here \(p\) is a probability.

\[\frac{dp}{dx} = p \left(1 - p \right)\] implies

\[p = \frac{e^x}{1 + e^x} + C\]

Code

plogisticC <- plogistic + 
  labs(subtitle = "p Grows Exponentially Until p Approaches 1.0",
       y = "probability")

ggplotly(plogisticC)