Advertisements
Advertisements
Question
If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0
Advertisements
Solution
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
RELATED QUESTIONS
If `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that" dy/dx = [-99x^2]/[101y^2]`
find dy/dx if x=e2t , y=`e^sqrtt`
If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and hence, find dy/dx if x=a cost, y=a sint
If x=at2, y= 2at , then find dy/dx.
If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`
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`
Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`
If x = cos t (3 – 2 cos2 t) and y = sin t (3 – 2 sin2 t), find the value of dx/dy at t =4/π.
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 equations, without eliminating the parameter, find `bb(dy/dx)`.
x = sin t, y = cos 2t
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = 4t, y = `4/y`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
`x = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = `a(cos t + log tan t/2)`, y = a sin t
If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.
x = a sec θ, y = b tan θ
If x = `sqrt(a^(sin^(-1)t))`, y = `sqrt(a^(cos^(-1)t))` show that `dy/dx = - y/x`.
If `x = acos^3t`, `y = asin^3 t`,
Show that `(dy)/(dx) =- (y/x)^(1/3)`
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find `dy/dx` when `theta = pi/3`
IF `y = e^(sin-1x) and z =e^(-cos-1x),` prove that `dy/dz = e^x//2`
The cost C of producing x articles is given as C = x3-16x2 + 47x. For what values of x, with the average cost is decreasing'?
x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`
x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ
If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at t" = pi/4) = "b"/"a"`
Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan–1x, when x ≠ 0
Form the point of intersection (P) of lines given by x2 – y2 – 2x + 2y = 0, points A, B, C, Dare taken on the lines at a distance of `2sqrt(2)` units to form a quadrilateral whose area is A1 and the area of the quadrilateral formed by joining the circumcentres of ΔPAB, ΔPBC, ΔPCD, ΔPDA is A2, then `A_1/A_2` equals
In the parametric equations \(x=f(t)\) and \(y=g(t)\), what is \(t\) called?
If required, in terms of which variables may the final answer be expressed instead of the parameter?
For \(x=a\cos^3\theta\) and \(y=a\sin^3\theta\), what are \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\)?
If \(x=a\cos t\) and \(y=a\sin t\), what are the derivatives with respect to \(t\)?
The formula \(\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) is based on which rule?
What should always be checked before using the main formula for the derivative of parametric functions?
