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`.
Advertisements
उत्तर
We have y = `e^(a cos^(-1)x)` ...(1)
Differentiating (1) both sides w.r.t. x, we get
`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))` ...(2)
Differentiating (2) both sides w.r.t. x, we get
`(d^2y)/(dx^2) = -a[(sqrt(1-x^2) dy/dx - y d/dx sqrt(1 - x^2))/((1-x^2))]`
`(d^2y)/(dx^2) = -a[(sqrt(1-x^2)dy/dx - y/(2sqrt(1-x^2)) * (-2x))/((1-x^2))]`
`(1 - x^2) (d^2y)/dx^2 = -a[-ay + (xy)/sqrt(1-x^2)]` ....[from (2)]
`(1 - x^2) (d^2y)/dx^2 = -a[-ay + x * ((-1)/a * dy/dx)]`
`(1 - x^2) (d^2y)/(dx^2) = a^2y + x dy/dx`
`(1 - x^2) (d^2y)/(dx^2) - x dy/dx - a^2y = 0`
APPEARS IN
संबंधित प्रश्न
if xx+xy+yx=ab, then find `dy/dx`.
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)`
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `(d^2y)/(dx^2)` , if y = log x
Differentiate : log (1 + x2) w.r.t. cot-1 x.
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
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)
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
Derivative of loge2 (logx) with respect to x is _______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`8^x/x^8`
`log (x + sqrt(x^2 + "a"))`
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
Evaluate:
`int log x dx`
Find the derivative of `y = log x + 1/x` with respect to x.
