Question

solve the following equation by changing the independent variable (1 + x²)² y'' + 2x(1 + x²) y' + 4y = 0.

Answer 1

We can solve this 2nd order ODE  by substituting for the variable  x.
           (1+x²)² y'' + 2 x (1+x²) y' + 4 y = 0       ---- (1)

Let  x = tan Ф,  dФ/dx = cos²Ф     ,    x = 0, when Ф = 0.
          dy/dx = dy/dФ * dФ/dx = dy/dФ * cos²Ф     ---(2)
          d²y/dx² = d/dФ (dy/dФ * cos²Ф)  * dФ/dx
                      = [ cos²Ф * d²y/dФ² - 2 cosФ sinФ* dy/dФ] * cos²Ф 
                      = cos⁴Ф * d²y/dФ² - 2 cos³Ф sinФ * dy/dФ   ---(3)

So (1) becomes:
         d²y/dФ² - 2 tanФ dy/dФ + 2 tanФ sec²Ф *dy/dФ * cos²Ф + 4 y = 0
         d²y/dФ² = - 4 y       --- (4)
This is the equation of motion for SHM... with ω = √4 = 2

    Let   y = A sin ωФ = A Sin2Ф         , Assuming Ф = 0 = Ф₀   for y = 0
         If the initial conditions are different we can have :  y = A sin (Ф+Ф₀).
     x = tanФ    So  cos2Ф = (1-x²)/(1+x²),   sin2Ф = 2x/(1+x²)

Solution:   y =  2 A x /(1+x²)          ---- (4)

We can also verify the solution by differentiating the above :

            y' = (1-x²)/(1+x²)               .    y'' = (x² - 3) 2x /(1+x²)³