Advertisements
Advertisements
प्रश्न
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Advertisements
उत्तर
log y = log (sin x) – x2
∴ log y = `log (sin x) – log e^(x^2)`
∴ log y = `log(sinx/e^(x^2))`
∴ y = `sinx/e^(x^2)`
∴ `e^(x^2).y` = sin x ...(1)
Differentiating both sides w.r.t. x, we get
`e^(x^2)."dy"/"dx" + y."d"/"dx"e^(x^2) = "d"/"dx"(sinx)`
∴ `e^(x^2)."dy"/"dx" + y.e^(x^2)."d"/"dx"(x^2)` = cos x
∴ `e^(x^2)."dy"/"dx" + y.e^(x^2) xx 2x` = cos x
∴ `e^(x^2)("dy"/"dx" + 2xy)` = cos x
Differentiating again w.r.t. x, we get
`e^(x^2)."d"/"dx"("dy"/"dx" + 2xy) + ("dy"/"dx" + 2xy)."d"/"dx"(e^(x^2)) = "d"/"dx"(cosx)`
∴ `e^(x^2)[(d^2y)/(dx^2) + 2(x"dy"/"dx" + y xx 1)] + ("dy"/"dx" + 2xy).e^(x^2)."d"/"dx"(x^2)` = – sin x
∴ `e^(x^2)[(d^2y)/(dx^2) + 2x"dy"/"dx" + y] + ("dy"/"dx" + 2xy).e^(x^2) xx 2x` = – sin x
∴ `e^(x^2)[(d^2y)/(dx^2) + 2x"dy"/"dx" + 2y + 2x"dy"/"dx" + 4x^2y]`
= `-e^(x^2).y` ...[By (1)]
∴ `(d^2y)/(dx^2) + 4x"dy"/"dx" + 4x^2y + 2y` = – y
∴ `(d^2y)/(dx^2) + 4x"dy"/"dx" + 4x^2y + 2y + y` = 0
∴ `(d^2y)/(dx^2) + 4x"dy"/"dx" + 4x^2y + 3y` = 0
∴ `(d^2y)/(dx^2) + 4x"dy"/"dx" + (4x^2 + 3)y` = 0.
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
2x + 3y = sin y
Find `bb(dy/dx)` in the following:
x3 + x2y + xy2 + y3 = 81
Is |sin x| differentiable? What about cos |x|?
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"`, if : x = `(t + 1/t)^a, y = a^(t+1/t)`, where a > 0, a ≠ 1, t ≠ 0.
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
Differentiate xx w.r.t. xsix.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x2 – 1)y2 + 4xy1 – y = 0.
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
Choose the correct option from the given alternatives :
If y = sin (2sin–1 x), then dx = ........
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)` = .........
Solve the following :
f(x) = –x, for – 2 ≤ x < 0
= 2x, for 0 ≤ x < 2
= `(18 - x)/(4)`, for 2 < x ≤ 7
g(x) = 6 – 3x, for 0 ≤ x < 2
= `(2x - 4)/(3)`, for 2 < x ≤ 7
Let u (x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v' (1) and w'(1). If it doesn't exist, then explain why?
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
If `x^7 * y^9 = (x + y)^16`, then show that `dy/dx = y/x`
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
State whether the following statement is True or False:
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))`
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
y = `e^(x3)`
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log (x + y) = log (xy) + a then show that, `dy/dx = (−y^2)/x^ 2`
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Find `dy/dx` if, `x = e^(3t), y = e^sqrtt`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
