Explanation:
(1) y = x e^(x²)
Take derivative with respect to x:
dy/dx = x (e^(x²) 2x) + e^(x²)
dy/dx = 2x² e^(x²) + e^(x²)
dy/dx = (2x² + 1) e^(x²)
Take derivative with respect to x again:
d²y/dx² = (2x² + 1) (e^(x²) 2x) + (4x) e^(x²)
d²y/dx² = (4x³ + 2x) e^(x²) + 4x e^(x²)
d²y/dx² = (4x³ + 6x) e^(x²)
Substitute:
d²y/dx² − 2x dy/dx − 4y
= (4x³ + 6x) e^(x²) − 2x (2x² + 1) e^(x²) − 4x e^(x²)
= 4x³ + 6x − 2x (2x² + 1) − 4x
= 4x³ + 6x − 4x³ − 2x − 4x
= 0
(2) y = sin⁻¹(√x)
sin y = √x
sin²y = x
Take derivative with respect to x:
2 sin y cos y dy/dx = 1
sin(2y) dy/dx = 1
dy/dx = csc(2y)
Take derivative with respect to x again:
d²y/dx² = -csc(2y) cot(2y) 2 dy/dx
d²y/dx² = -2 csc²(2y) cot(2y)
Substitute:
2x (1 − x) d²y/dx² + (1 − 2x) dy/dx
= 2 sin²y (1 − sin²y) (-2 csc²(2y) cot(2y)) + (1 − 2 sin²y) csc(2y)
Use power reduction formula:
= (1 − cos(2y)) (1 − ½ (1 − cos(2y))) (-2 csc²(2y) cot(2y)) + (1 − (1 − cos(2y))) csc(2y)
= (1 − cos(2y)) (1 − ½ + ½ cos(2y)) (-2 csc²(2y) cot(2y)) + cos(2y) csc(2y)
= (1 − cos(2y)) (½ + ½ cos(2y)) (-2 csc²(2y) cot(2y)) + cot(2y)
= (cos(2y) − 1) (1 + cos(2y)) csc²(2y) cot(2y) + cot(2y)
= (cos²(2y) − 1) csc²(2y) cot(2y) + cot(2y)
= -sin²(2y) csc²(2y) cot(2y) + cot(2y)
= -cot(2y) + cot(2y)
= 0