If cos ⁻¹( y/2 ) = log ₑ( x/5 )⁵, |y| < 2, then :

Solution

Maths sir · 4 steps

Turn the equation into y = 2 cos θ, then differentiate xy' once more instead of y'.

1. Make y explicit. Write logₑ(x/5)⁵ = 5 ln x/5 and call it θ. Then cos⁻¹ y/2 = θ gives y = 2 cos θ, with dθ/dx = 5/x.
2. First derivative. y' = -2 sin θ × 5/x, so xy' = -10 sin θ. Keep it in this form. Clearing the x makes the next derivative much easier.

   d/dx(xy') = y' + xy'' = -10 cos θ × 5/x = -(50 cos θ)/x

3. Replace cos θ. Multiply through by x: xy' + x²y'' = -50 cos θ. Since 2 cos θ = y, the right side is -25y.
4. So. x²y'' + xy' + 25y = 0, which is option (d).

The idea to keep. When y depends on ln x, first find xy', then differentiate that. You get x²y'' + xy' directly, and it equals d²y/dt² with t = ln x.

Answer from the NTA final answer key. Solution by PrepWiser.