Solving ODEs (solve_ivp) in SciPy - Mini Project: Build & Apply

Solving ODEs with solve_ivp

📖 Scenario: Imagine you are studying how a simple population grows over time. The population changes according to a rule that depends on its current size. This is a common problem in biology and ecology.

🎯 Goal: You will write a program to solve a simple ordinary differential equation (ODE) that models population growth using solve_ivp from the scipy library. You will define the growth rule, set the time range, solve the ODE, and then display the population values over time.

📋 What You'll Learn

Use the solve_ivp function from scipy.integrate to solve the ODE

Define the ODE function with the correct signature

Set the initial population value and time span

Extract and print the solution values

💡 Why This Matters

🌍 Real World

Solving ODEs is essential in fields like biology, physics, and engineering to model how things change over time, such as populations, chemical reactions, or mechanical systems.

💼 Career

Data scientists and analysts often need to model dynamic systems and predict future behavior, making ODE solving a useful skill in simulation and forecasting tasks.

1

Define the population growth function

Import solve_ivp from scipy.integrate and define a function called population_growth that takes t and y as inputs and returns 0.1 * y. This models a growth rate of 10%.

SciPy

# Import solve_ivp and define the population_growth function
# Your code here

Need a hint?

Remember, the function for solve_ivp must take time t and state y as inputs and return the rate of change.

2

Set the initial population and time span

Create a variable called y0 and set it to [100] to represent the initial population. Create a variable called t_span and set it to the tuple (0, 50) to represent the time from 0 to 50.

SciPy

from scipy.integrate import solve_ivp

def population_growth(t, y):
    return 0.1 * y

# Set initial population y0 and time span t_span
# Your code here

Need a hint?

The initial population should be a list or array with one value, and the time span is a tuple with start and end times.

3

Solve the ODE using solve_ivp

Use solve_ivp with the function population_growth, the time span t_span, and the initial value y0. Save the result in a variable called solution.

SciPy

from scipy.integrate import solve_ivp

def population_growth(t, y):
    return 0.1 * y


y0 = [100]
t_span = (0, 50)

# Solve the ODE and save the result in solution
# Your code here

Need a hint?

Call solve_ivp with the function, time span, and initial values in that order.

4

Print the population values over time

Print the array solution.t which contains the time points, and then print solution.y[0] which contains the population values at those times.

SciPy

from scipy.integrate import solve_ivp

def population_growth(t, y):
    return 0.1 * y


y0 = [100]
t_span = (0, 50)

solution = solve_ivp(population_growth, t_span, y0)

# Print the time points and population values
# Your code here

Need a hint?

Use print(solution.t) and print(solution.y[0]) to show the time points and population values.