library(deSolve)
library(ggplot2)

# Define the differential equation function
bank_account <- function(time, state, parameters) {
  with(as.list(c(state, parameters)), {
    # Define the differential equation
    dBalance <- InterestRate * Balance
    
    # Return the derivatives
    return(list(c(dBalance)))
  })
}

# Set the initial conditions and parameters
initial_balance <- 1000  # Initial balance of the bank account
interest_rate <- 0.05    # Annual interest rate

# Create the parameter list
parameters <- list(
  InterestRate = interest_rate
)

# Create the initial state vector
initial_state <- c(
  Balance = initial_balance
)

# Set the time span for the simulation
time_span <- c(0, 10)  # Simulation for 10 years

# Solve the differential equation
solution <- ode(
  y = initial_state,
  times = time_span,
  func = bank_account,
  parms = parameters
)

# Access the results
time <- solution[, 1]
balance <- solution[, 2]

# Create a data frame with the time and balance
data <- data.frame(time = time, balance = balance)

# Create the plot using ggplot2
plot <- ggplot(data, aes(x = time, y = balance)) +
  geom_line() +
  labs(x = "Time", y = "Bank Account Balance") +
  theme_minimal()

# Save the plot as a PDF file
ggsave("bank_account_plot.pdf", plot, width = 8, height = 6)