Advertisements
Advertisements
प्रश्न
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
Advertisements
उत्तर
y = sin (m cos–1x)
∴ sin–1 y = m cos–1 x
Differentiating both sides w.r.t. x, we get
`(1)/sqrt(1 - y^2)."dy"/"dx" = m xx (-1)/sqrt(1 - x^2)`
∴ `sqrt(1 - x^2)."dy"/"dx" = -msqrt(1 - y^2)`
∴ `(1 - x^2)(dy/dx)^2 = m^2(1 - y^2)`
∴ `(1 - x^2)(dy/dx)^2` = m2 – m2y2
Differentiating both sides w.r.t. x, we get
`(1 - x^2)."d"/"dx"(dy/dx)^2 + (dy/dx)^2."d"/"dx"(1 - x^2) = 0 - m^2."d"/"dx"(y^2)`
∴ `(1 - x^2).2"dy"/"dx".(d^2y)/(dx^2) - 2x(dy/dx)^2 = -2m^2y"dy"/"dx"`
Cancelling `2"dy"/"dx"` throughtout, we get
`(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx"` = – m2y
∴ `( 1- x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
APPEARS IN
संबंधित प्रश्न
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `bb(dy/dx)` in the following:
2x + 3y = sin y
Find `bb(dy/dx)` in the following:
x2 + xy + y2 = 100
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
Is |sin x| differentiable? What about cos |x|?
Find `dy/dx if x^3 + y^2 + xy = 7`
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Differentiate e4x + 5 w.r..t.e3x
Differentiate tan-1 (cot 2x) w.r.t.x.
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.
Find the nth derivative of the following : `(1)/(3x - 5)`
Choose the correct option from the given alternatives :
If y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
Choose the correct option from the given alternatives :
If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
Differentiate the following w.r.t. x:
`tan^-1(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
If x sin (a + y) + sin a . cos (a + y) = 0, then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
Find `dy/dx if, x= e^(3t), y = e^sqrtt`
`"If" log(x+y) = log(xy)+a "then show that", dy/dx=(-y^2)/x^2`
Find `dy/dx` if , x = `e^(3t), y = e^(sqrtt)`
Find `dy/dx if , x = e^(3t) , y = e^sqrtt`
Find `dy / dx` if, x = `e^(3t), y = e^sqrt t`
Solve the following.
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if, x = e3t, y = `e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx=(-y^2)/x^2`
