Advertisements
Advertisements
Question
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
Advertisements
Solution
y = 500e7x + 600e–7x ...(1)
On differentiating with respect to x,
`dy/dx = d/dx (500 e^(7x) + 600 e^(- 7x))`
= `500 d/dx e^(7x) + 600 d/dx e^(- 7x)`
= `500 e^(7x) d/dx (7x) + 600 e^(- 7x) d/dx (-7x)`
= 500e7x . 7 + 600e–7x. (−7)
= 3500e7x − 4200e−7x
Differentiating again with respect to x,
`(d^2 y)/dx^2 = d/dx [3500e^(7x) − 4200e^(−7x)]`
= `3500 d/dx e^(7x) - 4200 d/dx e^(- 7x)`
= `3500e^(7x) d/dx (7x) - 4200e^(- 7x) d/dx (- 7x)`
= 3500e7x . 7 − 4200e−7x . (−7)
= 24500e7x − 29400e−7x
= 500 × 49e7x + 600 × 49e−7x
= 49(500e7x + 600e−7x)
= 49 y ...[From equation (1)]
∴ `(d^2y)/dx^2` = 49y
APPEARS IN
RELATED QUESTIONS
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`
Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
log x
Find the second order derivative of the function.
ex sin 5x
Find the second order derivative of the function.
tan–1 x
Find the second order derivative of the function.
sin (log x)
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
Find `("d"^2"y")/"dx"^2`, if y = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
Find `("d"^2"y")/"dx"^2`, if y = log (x).
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
`sin xy + x/y` = x2 – y
tan–1(x2 + y2) = a
(x2 + y2)2 = xy
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/(dx^2)` if, y = `e^((2x+1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`
