Question

If y = f(x), f'(0) = f(0) = 1 and if y = f(x) satisfies `(d^2y)/(dx^2) + (dy)/(dx)` = x, then the value of [f(1)] is ______ (where [.] denotes greatest integer function)

पर्याय

• 0.00

• 1.00

• 2.00

• 3.00

Answer 1

If y = f(x), f'(0) = f(0) = 1 and if y = f(x) satisfies `(d^2y)/(dx^2) + (dy)/(dx)` = x, then the value of [f(1)] is 1.00 (where [.] denotes greatest integer function)

Explanation:

`(d^2y)/(dx^2) + (dy)/(dx)` = x

Let `(dy)/(dx)` = t

⇒ `(d^2y)/(dx^2) = (dt)/(dx)`

⇒ `(dt)/(dx) + t` = x

⇒ te^x = `intxe^x dx + c`

⇒ te^x = xe^x – e^x + c

⇒ `(dy)/(dx)` = x – 1 + ce^–x 

y'(0) = 1

⇒ c = 2

⇒ `(dy)/(dx)` = 2e^–x + x – 1

⇒ y = `-2e^-x + x^2/2 - x + c`

⇒ y(0) = 1

⇒ c = 3

⇒ y(1) = `-2/e + 1/2 - 1 + 3`

⇒ [y(1)] = 1