Variation of Parameters Step

Problem

For y'' + y = f(x) with fundamental solutions y1 = cos(x) and y2 = sin(x), variation of parameters gives u1' = -y2*f / W and u2' = y1*f / W, where W = 1.

Given the value f_value = f(x) at the point x, return [u1', u2'] = [-sin(x)*f_value, cos(x)*f_value].

Hint

The Wronskian of cos and sin is 1, which simplifies the formulas.

Test cases

case 1

in: [1,0]

expected: [0,1]

case 2

in: [1,1.5707963267948966]

expected: [-1,0]

case 3

in: [2,3.141592653589793]

expected: [0,-2]

case 4

in: [3,0.5235987755982988]

expected: [-1.5,2.598076211353316]

Starter codePython

def variation_step(f_value, x):
    # Return [u1_prime, u2_prime]
    # Your code here
    pass

4 test cases · run your code against them in the editor