Advertisements
Advertisements
प्रश्न
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
उत्तर
y = A cos (log x) + B sin (log x) ...(1)
Differentiating both sides w.r.t. x, we get
`"dy"/"dx" = "A""d"/"dx"[cos(logx)] + "B""d"/"dx"[sin(log x)]`
= `"A"[-sin (logx)]."d"/"dx"(logx) + "B"cos(logx)."d"/"dx"(logx)`
= `"A"sin(logx) xx (1)/x "B"cos(logx) xx(1)/x`
∴ `x"d"/"dx"(dy/dx) + "dy"/"dx"."d"/"dx"(x) = -"A""d"/"dx"[sin(logx)] +"B""d"/"dx"[cos(logx)]`
∴ `x(d^2y)/(dx2) + "dy"/"dx" xx 1 = -"A"cos(logx)."d"/"dx"(logx) + "B"[-sin(logx)]."d"/"dx"(logx)`
∴ xy2 + y1 = `-"A"cos(logx) xx(1)/x - "B"sin(logx) xx (1)/x`
∴ x2y2 + xy1 = – [A cos (log x) + B sin (log x)] ...[By (1)]
∴ x2y2 + xy1 + y = 0.
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `dy/dx` for the function given in the question:
yx = xy
Find `dy/dx` for the function given in the question:
`xy = e^((x – y))`
Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ′(1).
Differentiate w.r.t. x the function:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
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`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `dy/dx` if y = xx + 5x
Find `(d^2y)/(dx^2)` , if y = log x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
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`.
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.logx
Find the second order derivatives of the following : log(logx)
Find the nth derivative of the following : log (ax + b)
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`d/dx(x^{sinx})` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
