If y^(1/4) + y^(- 1/4) = 2x, and
(x² - 1)d²y/dx² + α xdy/dx + β y = 0, then | α - β | is equal to __________.

Solution

Maths sir · 4 steps

Solve for y^(1/4) first, then square the first-derivative relation to clear the root.

1. Find y. Put t = y^(1/4), so t + 1/t = 2x, which gives t² - 2xt + 1 = 0. Then t = x ± √(x²-1) and y = (x ± √(x²-1))⁴.
2. First derivative. For t = x + √(x²-1), t' = 1 + x/√(x²-1) = t/√(x²-1). So y' = 4t³ t' = 4y/√(x²-1). The minus root gives y' = -4y/√(x²-1), and the next step is the same for both.
3. Square and differentiate. Squaring gives (x²-1)(y')² = 16y². Differentiate both sides: 2x(y')² + 2(x²-1)y'y'' = 32yy'. Divide by 2y'.

   (x²-1)d²y/dx² + xdy/dx - 16y = 0

4. So. Comparing, α = 1 and β = -16, so |α - β| = |1 + 16| = 17.

The common slip: 15. Taking β = 16 instead of -16 gives |1 - 16| = 15, not 17. The 16y term moves to the left side with a minus sign.

The idea to keep. When y' comes with a square root like √(x²-1), square the relation before differentiating again. The root vanishes and the second-order equation drops out.

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