Advertisements
Advertisements
प्रश्न
If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0
उत्तर
Given that: x = sin t and y = sin pt
Differentiating both the parametric functions w.r.t. t
`"dx"/"dt"` = cos t and `"dy"/"dt"` = cos pt. p = p cos pt
`"dy"/"dx" = ("dy"/"dt")/("dx"/"dt")`
= `("p" cos "pt")/cos "t"`
∴ `"dy"/"dx" = ("p" cos "pt")/cos"t"`
Again differentiating w.r.t. x,
`"d"/"dx"("dy"/"dx") = "p"*"d"/"dx"(cos"pt"/cos"t")`
⇒ `("d"^2y)/("dx"^2) = "p"[(cos "t" * "d"/"dx" (cos "pt") - cos "pt" * "d"/"dx" (cos "t"))/(cos^2"t")]`
= `"p"[(cos"t"(- sin "pt") * "p" "dt"/"dx" - cos "pt"(- sin "t") * "dt"/"dx")/(cos^2"t")]`
= `"p"[(-"p" cos "t" sin "pt" + cos "pt" sin "t")/(cos^2"t")]"dt"/"dx"`
= `"p"[(-"p" cos "t" sin "pt" + cos "pt" sin "t")/(cos^2"t")]* 1/cos"t"`
= `"p"[(-"p" cos "t" sin "pt" + cos "pt" sin "t")/cos^3"t"]`
Now we have to prove that
`(1 - x^2) * ("d"^2y)/("dx"^2) - x * "dy"/"dx" + "p"^2y` = 0
L.H.S. = `(1 - x^2) ["p"((-"p" cos "t" sin "pt" + cos "pt" sin "t")/cos^3"t")] - x."p" (cos "pt")/cos"t" + "p"^2y`
⇒ `(1 - sin^2"t") ["p"((-"p" cos "t" sin "pt" + cos "pt" sin "t")/cos^3"t")] - ("p" sin "t" * cos "pt")/cos"t" + "p"^2 * sin "pt"`
⇒ `cos^2"t" [(-"p"^2 cos "t" sin "pt" + cos "pt" sin "t")/(cos^3"t")] - ("p" sin "t" * cos "pt")/cos"t" + "p"^2 * sin "pt"`
⇒ `(-"p"^2 cos "t" sin "pt" + "p" cos "pt" sin "t")/cos "t" - ("p" sin "t" cos "pt")/cos"t" + "p"^2 sin "pt"`
⇒ `(-"p"^2 cos "t" sin "pt" + "p" cos "pt" sin "t" - "p" sin "t" cos "pt" + "p"^2 sin "pt" cos "t")/cos "t"`
⇒ `0/cos"t"` = 0 = R.H.S.
Hence proved.
APPEARS IN
संबंधित प्रश्न
If `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that" dy/dx = [-99x^2]/[101y^2]`
If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`
If y =1 - cos θ , x = 1 - sin θ , then ` dy/dx at " "θ =pi/4` is ________
If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `
If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that `dy/dx=β/αtan t`
If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`
If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find `dy/dx `
Derivatives of tan3θ with respect to sec3θ at θ=π/3 is
(A)` 3/2`
(B) `sqrt3/2`
(C) `1/2`
(D) `-sqrt3/2`
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`
`x = 2at^2, y = at^4`
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
x = a cos θ, y = b cos θ
If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx.`
x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find `dy/dx` when `theta = pi/3`
If X = f(t) and Y = g(t) Are Differentiable Functions of t , then prove that y is a differentiable function of x and
`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`
Hence find `"dy"/"dx"` if x = a cos2 t and y = a sin2 t.
If y = sin -1 `((8x)/(1 + 16x^2))`, find `(dy)/(dx)`
Evaluate : `int (sec^2 x)/(tan^2 x + 4)` dx
sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`
x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`
If x = ecos2t and y = esin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`
If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at t" = pi/4) = "b"/"a"`
If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`
Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan–1x, when x ≠ 0
If `"x = a sin" theta "and y = b cos" theta, "then" ("d"^2 "y")/"dx"^2` is equal to ____________.
