ODE solver methods (RK45, BDF) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the ODE solver function from scipy.

SciPy

from scipy.integrate import [1]

Drag options to blanks, or click blank then click option'

Aodeint

Bode

Csolve_ivp

Dintegrate

Attempts:

3 left

💡 Hint

Common Mistakes

Using odeint which is older and less flexible.

Trying to import a non-existent function.

The solve_ivp function is the modern ODE solver in scipy that supports methods like RK45 and BDF.

2fill in blank

medium

Complete the code to define the ODE function for dy/dt = -2y.

SciPy

def dydt(t, y):
    return [1] * y

Drag options to blanks, or click blank then click option'

A2

B-2

C-y

Dy

Attempts:

3 left

💡 Hint

Common Mistakes

Returning just -y instead of -2*y.

Returning a positive 2 instead of negative.

The ODE is dy/dt = -2y, so the function returns -2 times y.

3fill in blank

hard

Fix the error in the code to solve the ODE using RK45 method.

SciPy

sol = solve_ivp(dydt, [0, 5], [1], method=[1])

Drag options to blanks, or click blank then click option'

A"bdf"

B"euler"

C"rk45"

D"RK45"

Attempts:

3 left

💡 Hint

Common Mistakes

Using lowercase 'rk45' which causes an error.

Using unsupported method names like 'euler'.

The method name is case-sensitive and should be "RK45" with uppercase letters.

4fill in blank

hard

Fill both blanks to solve the ODE using BDF method and print the solution at t=5.

SciPy

sol = solve_ivp(dydt, [0, [1]], [1], method=[2])
print(sol.y[0, -1])

Drag options to blanks, or click blank then click option'

A5

B10

C"RK45"

D"BDF"

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong end time like 10.

Using lowercase method names.

The time interval ends at 5, and the method for stiff problems is "BDF".

5fill in blank

hard

Fill all three blanks to create a dictionary of solutions at specific times using RK45 method.

SciPy

times = [0, 1, 2, 3]
sol = solve_ivp(dydt, [0, 3], [1], method=[1], t_eval=[2])
results = {t: y for t, y in zip(sol.t, sol.y[0]) if t [3] 0}

Drag options to blanks, or click blank then click option'

A"RK45"

B[0, 1, 2, 3]

C>=

D<

Attempts:

3 left

💡 Hint

Common Mistakes

Not passing t_eval to get solution at specific times.

Using wrong comparison operator in dictionary comprehension.

Use "RK45" as method, provide t_eval as the list of times, and filter times greater or equal to 0.

Practice

1. Which scipy ODE solver method is best suited for stiff problems?

easy

A. BDF

B. RK45

C. Euler

D. Midpoint

Solution

1. Step 1: Understand stiff problems

   Stiff problems require methods that handle rapid changes without instability.

2. Step 2: Match solver to problem type

   BDF (Backward Differentiation Formula) is designed for stiff problems, unlike RK45 which is for general cases.

3. Final Answer:

   BDF -> Option A

4. Quick Check:

   Stiff problems = BDF [OK]

Hint: BDF is for stiff problems, RK45 for general ones [OK]

Common Mistakes:

• Confusing RK45 as best for stiff problems
• Thinking Euler method is in solve_ivp
• Assuming Midpoint is a scipy method

2. Which of the following is the correct way to specify the RK45 method in solve_ivp?

easy

A. method='RungeKutta45'

B. method='rk45'

C. method='RK_45'

D. method='RK45'

Solution

1. Step 1: Check method parameter syntax

   The solve_ivp function expects method names as strings exactly matching documented names.

2. Step 2: Verify correct spelling and case

   The correct method name is 'RK45' with uppercase letters and no underscores or spaces.

3. Final Answer:

   method='RK45' -> Option D

4. Quick Check:

   Exact method name = 'RK45' [OK]

Hint: Use exact method names with correct case in solve_ivp [OK]

Common Mistakes:

• Using lowercase 'rk45' instead of 'RK45'
• Adding underscores like 'RK_45'
• Using descriptive names not accepted by solve_ivp

3. What will be the output of this code snippet?

from scipy.integrate import solve_ivp
import numpy as np

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

sol = solve_ivp(fun, [0, 2], [1], method='RK45', t_eval=[0, 1, 2])
print(sol.y[0])

medium

A. [1. 0.13533528 0.01831564]

B. [1. 0.5 0.25]

C. [1. 2. 4.]

D. [0. 0. 0.]

Solution

1. Step 1: Understand the ODE and solution

   The ODE is dy/dt = -2y with initial y(0)=1, which has exact solution y(t) = exp(-2t).

2. Step 2: Calculate values at t=0,1,2

   y(0)=1, y(1)=exp(-2) ≈ 0.1353, y(2)=exp(-4) ≈ 0.0183.

3. Final Answer:

   [1. 0.13533528 0.01831564] -> Option A

4. Quick Check:

   exp(-2t) at t=0,1,2 = [1. 0.13533528 0.01831564] [OK]

Hint: Use exact solution exp(-2t) to check output array [OK]

Common Mistakes:

• Confusing exponential decay with linear decay
• Misreading t_eval points
• Assuming RK45 returns integers

4. Identify the error in this code using BDF method:

from scipy.integrate import solve_ivp

def f(t, y):
    return y - t**2 + 1

sol = solve_ivp(f, [0, 2], 0.5, method='BDF', t_eval=[0, 1, 2])
print(sol.y)

medium

A. Function f must take only one argument

B. Initial condition y0 should be a list or array, not a float

C. Method 'BDF' is not supported in solve_ivp

D. t_eval must be a numpy array, not a list

Solution

1. Step 1: Check initial condition format

   The initial condition y0 must be an array-like object, even if single value, e.g., [0.5].

2. Step 2: Verify given y0 type

   Here y0 is given as 0.5 (float), which is not array-like and causes error.

3. Step 3: Confirm other parameters

   Function f has correct signature, method 'BDF' is valid, t_eval can be list or array.

4. Final Answer:

   Initial condition y0 should be a list or array, not a float -> Option B

5. Quick Check:

   y0 must be array-like [OK]

Hint: Always pass y0 as list or array, not single float [OK]

Common Mistakes:

• Passing y0 as float instead of list
• Thinking BDF is unsupported
• Believing t_eval must be numpy array

5. You want to solve a stiff ODE system and compare solutions from RK45 and BDF methods. Which approach correctly compares their results at specific times?

hard

A. Use only RK45 since BDF is slow and ignore BDF results

B. Run RK45 without t_eval and BDF with t_eval, then compare sol.t arrays

C. Use solve_ivp with t_eval for both methods, then compare sol.y arrays at those times

D. Compare the initial conditions only, since methods differ internally

Solution

1. Step 1: Understand comparison requirements

   To compare solutions, evaluate both methods at the same time points using t_eval.

2. Step 2: Use solve_ivp with t_eval for both

   Passing t_eval ensures outputs have matching time points, so sol.y arrays can be directly compared.

3. Step 3: Reject incorrect approaches

   Not using t_eval leads to different time points; ignoring BDF or comparing only initial conditions misses solution differences.

4. Final Answer:

   Use solve_ivp with t_eval for both methods, then compare sol.y arrays at those times -> Option C

5. Quick Check:

   Same t_eval for both = valid comparison [OK]

Hint: Use t_eval to get matching time points for comparison [OK]

Common Mistakes:

• Not using t_eval for both methods
• Comparing sol.t arrays instead of sol.y
• Ignoring BDF results due to speed concerns