ODE solver methods (RK45, BDF) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of RK45 solver on a simple ODE

What is the approximate value of y(2) when solving dy/dt = -2y with y(0) = 1 using RK45 method?

SciPy

from scipy.integrate import solve_ivp

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

sol = solve_ivp(fun, [0, 2], [1], method='RK45')
result = sol.y[0, -1]
print(round(result, 3))

A 0.135

B 0.368

C 0.018

D 0.500

💡 Hint

Recall the exact solution is y(t) = exp(-2t). RK45 approximates this numerically.

The exact solution at t=2 is exp(-4) ≈ 0.0183. RK45 with default tolerances and step sizes accurately approximates this to 0.018 when rounded to 3 decimals.

❓ Predict Output

intermediate

Output difference between RK45 and BDF on stiff ODE

Given the stiff ODE dy/dt = -1000y + 3000 - 2000 exp(-t), y(0)=0, what is the approximate value of y(0.1) using BDF method?

SciPy

import numpy as np
from scipy.integrate import solve_ivp

def stiff(t, y):
    return -1000*y + 3000 - 2000*np.exp(-t)

sol = solve_ivp(stiff, [0, 0.1], [0], method='BDF')
result = sol.y[0, -1]
print(round(result, 2))

A 0.00

B 1.19

C 0.10

D 1.00

💡 Hint

BDF is good for stiff problems and will quickly approach the steady state.

The stiff ODE solution at t=0.1 is close to 1.19 due to rapid initial change toward the quasi-equilibrium y ≈ 3 - 2*exp(-t). BDF handles stiffness well and produces this value.

🔧 Debug

advanced

Identify the error in ODE solver usage

What error will this code raise when trying to solve dy/dt = y with RK45 method? from scipy.integrate import solve_ivp def fun(t, y): return y sol = solve_ivp(fun, [0, 1], 1, method='RK45')

SciPy

from scipy.integrate import solve_ivp

def fun(t, y):
    return y

sol = solve_ivp(fun, [0, 1], 1, method='RK45')

A TypeError: 'int' object is not iterable

B AttributeError: 'int' object has no attribute 'shape'

C No error, runs successfully

D ValueError: y0 must be array-like

💡 Hint

Check the type of initial condition y0 parameter.

solve_ivp expects y0 as an array-like object (list or array). Passing an int causes ValueError.

❓ data_output

advanced

Number of time points returned by solve_ivp with dense output

Using solve_ivp with method='RK45' and dense_output=True on dy/dt = -y, y(0)=1 over [0,5], how many time points are in sol.t?

SciPy

from scipy.integrate import solve_ivp

def fun(t, y):
    return -y

sol = solve_ivp(fun, [0, 5], [1], method='RK45', dense_output=True)
print(len(sol.t))

A Varies depending on solver steps

B 100

C 50

D 10

💡 Hint

RK45 adapts step size, so number of points is not fixed.

The number of time points in sol.t depends on adaptive step size chosen by RK45, so it varies per run and problem.

🚀 Application

expert

Choosing solver method for a stiff system

You have a system of ODEs modeling chemical reactions with very fast and slow dynamics mixed. Which solver method is best to use in solve_ivp for stable and efficient results?

A BDF

B RK23

C Euler explicit

D RK45

💡 Hint

Stiff systems require implicit solvers for stability.

BDF is an implicit solver designed for stiff problems, providing stability and efficiency for mixed fast/slow dynamics.