Question

Let y = y(x), x > 1, be the solution of the differential equation `(x - 1)(dy)/(dx) + 2xy = 1/(x - 1)`, with y(2) = `(1 + e^4)/(2e^4)`. If y(3) = `(e^α + 1)/(βe^α)`, then the value of α + β is equal to ______.

Options

• 12

• 13

• 14

• 15

Answer 1

Let y = y(x), x > 1, be the solution of the differential equation `(x - 1)(dy)/(dx) + 2xy = 1/(x - 1)`, with y(2) = `(1 + e^4)/(2e^4)`. If y(3) = `(e^α + 1)/(βe^α)`, then the value of α + β is equal to 14.

Explanation:

`(x - 1)(dy)/(dx) + 2xy = 1/(x - 1)`

It is linear differential equation.

⇒ `(dy)/(dx) + (2x)/(x - 1)y = 1/((x - 1)^2`

Comparing above equation with `(dy)/(dx) + py` = Q, we get

P = `(2x)/(x - 1)y` and Q = `1/(x - 1)^2`

Now, I.F. = `e^(intP.dx)`

⇒ I.F. = `e^(int(2x)/(x - 1)dx`

⇒ I.F. = `e^(int(2 + 2/(x - 1))dx`

⇒ I.F. = `e^(2x + 2  "In"  |x - 1|)`

⇒ I.F. = `e^(2x).e^("In"(x - 1)^2`

⇒ I.F. = `(x - 1)^2e^(2x)`

So, the solution of a given differential equation is given by

y(I.F.) = `int"Q".("I"."F".)dx`

⇒ ye^2x(x – 1)² = `int1/(x - 1)^2 e^(2x) (x - 1)^2dx`

⇒ ye^2x(x – 1)² = `inte^(2x)dx`

⇒ ye^2x(x – 1)² = `e^(2x)/2 + "C"`

Put x = 2 in above equation, we get

y(2)e⁴ = `e^4/2 + "C"`

⇒ C = `(1 + e^4)/(2e^4).e^4 - e^4/2`  ......`{∵ y(2) = (1 + e^4)/(2e^4)}`

⇒ C = `1/2`

⇒ ye^2x(x – 1)² = `e^(2x)/2 + 1/2`

Put x = 3 in above equation, we get

y(3)e⁶4 = `e^6/2 + 1/2`

⇒  `(e^α + 1)/(βe^α) = (e^6 + 1)/(8e^6)`

⇒ α = 6, β = 8

∴ α + β = 14