Question

What will be the form of the equation after eliminating a and b from the equation y = a sin^-1x + b cos^-1x?

(a) (1 – x^2)y” + xy’ = 0

(b) (1 – x^2)y” – xy’ = 0

(c) (1 + x^2)y” – xy’ = 0

(d) (1 + x^2)y” + xy’ = 0

The question was asked in unit test.

My question is based upon Second Order Derivative in section Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct option is (b) (1 – x^2)y” – xy’ = 0

The best I can explain: We have, y = a sin^-1x + b cos^-1x

Differentiating both sides with respect to x we get,

dy/dx = 12∫ e^2xcos3x dx – 5∫ e^2x sin3x dx

y’ = a* 1/√(1 – x^2) + b * (-1/√(1 – x^2))

or, y’√(1 – x^2) = a – b

now, squaring both sides,

or, y’^2(1 – x^2) = (a – b)^2

Differentiating again with respect to x we get,

(1 – x^2) * 2y’y” + y’^2 . (-2x) = 0

As, y’ ≠ 0,

(1 – x^2)y” – xy’ = 0