Advertisements
Advertisements
प्रश्न
If `y = e^(acos^(-1)x)`, -1 <= x <= 1 show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`
उत्तर
y = `e^(a cos^(-1)x)`
On differentiating with respect to x,
`dy/dx = e^(a cos^(-1)x) d/dx a cos^-1 x`
`= e^(a cos^(-1)x) (- a)/sqrt(1 - x^2) = (- ay)/(sqrt(1 - x^2))`
On multiplying by `sqrt(1 - x^2)`,
`=> sqrt(1 - x^2) dy/dx` = - ay
On squaring,
`(dy/dx)^2 (1 - x^2) = a^2y^2`
Differentiating again with respect to x,
`=> 2 (dy/dx) (d^2y)/dx^2 (1 - x^2) + (dy/dx)^2 (- 2x) = 2a^2 y dy/dx`
Dividing by `2 dy/dx`,
`(d^2y)/dx^2 (1 - x^2) - x dy/dx = a^2 y`
Hence, `(1 - x^2) (d^2y)/dx^2 - x(dy/dx) - a^2 y = 0`
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
`x^x - 2^(sin x)`
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ′(1).
Evaluate
`int 1/(16 - 9x^2) dx`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If y = (log x)x + xlog x, find `"dy"/"dx".`
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
Find the second order derivatives of the following : x3.logx
Find the nth derivative of the following : log (ax + b)
Find the nth derivative of the following : log (2x + 3)
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`log (x + sqrt(x^2 + "a"))`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
