ODE solver methods (RK45, BDF) in SciPy - Mini Project: Build & Apply

Solving a Simple ODE with RK45 and BDF Methods

📖 Scenario: Imagine you want to understand how a simple system changes over time, like how the temperature cools down or how a population grows. We can use math equations called differential equations to describe these changes. To find out the values at different times, we use special tools called ODE solvers.

🎯 Goal: You will learn how to solve a simple ordinary differential equation (ODE) using two popular methods: RK45 and BDF. You will write code to set up the equation, choose the solver method, solve the equation, and see the results.

📋 What You'll Learn

Use the scipy.integrate.solve_ivp function to solve ODEs

Define a function for the differential equation

Set up initial conditions and time points

Solve the ODE using both RK45 and BDF methods

Print the solution values at the final time point

💡 Why This Matters

🌍 Real World

ODE solvers like RK45 and BDF are used in physics, biology, engineering, and finance to model how things change over time, such as chemical reactions, population growth, or mechanical systems.

💼 Career

Understanding how to solve ODEs with different methods is important for data scientists and engineers working on simulations, modeling, and predictive analytics.

1

Define the ODE function and initial conditions

Create a function called ode_function that takes t and y as inputs and returns -2 * y. Then create a variable called y0 and set it to [1] as the initial value. Finally, create a variable called t_span and set it to the tuple (0, 5) representing the time interval.

SciPy

# Your code here

Need a hint?

The function should return the rate of change at time t and value y. The initial value y0 is a list with one number. The time span is a tuple with start and end times.

2

Set the solver method to RK45

Create a variable called method and set it to the string 'RK45' to choose the Runge-Kutta 45 solver method.

SciPy

def ode_function(t, y):
    return -2 * y

y0 = [1]
t_span = (0, 5)

# Your code here

Need a hint?

Just assign the string 'RK45' to the variable method.

3

Solve the ODE using solve_ivp with the chosen method

Use solve_ivp from scipy.integrate to solve the ODE. Call it with ode_function, t_span, y0, and method=method. Save the result in a variable called solution.

SciPy

def ode_function(t, y):
    return -2 * y

y0 = [1]
t_span = (0, 5)

method = 'RK45'

# Your code here

Need a hint?

Remember to import solve_ivp from scipy.integrate before using it.

4

Print the solution value at the final time point

Print the last value of solution.y[0] using print(). This shows the solution at the end of the time interval.

SciPy

from scipy.integrate import solve_ivp

def ode_function(t, y):
    return -2 * y

y0 = [1]
t_span = (0, 5)

method = 'RK45'

solution = solve_ivp(ode_function, t_span, y0, method=method)

# Your code here

Need a hint?

The output should be close to 0.0067, which is the value of e^{-10}.