Question

Let y = y(x) be the solution curve of the differential equation `(dy)/(dx) + ((2x^2 + 11x + 13)/(x^3 + 6x^2 + 11x + 6)) y = ((x + 3))/(x + 1), x > - 1`, which passes through the point (0, 1). Then y(1) is equal to ______.

Options

• `1/2`

• `3/2`

• `5/2`

• `7/2`

Answer 1

Let y = y(x) be the solution curve of the differential equation `(dy)/(dx) + ((2x^2 + 11x + 13)/(x^3 + 6x^2 + 11x + 6)) y = ((x + 3))/(x + 1), x > - 1`, which passes through the point (0, 1). Then y(1) is equal to `underlinebb(3/2)`.

Explanation:

Given differential equation is

`(dy)/(dx) + ((2x^2 + 11x + 13)/(x^3 + 6x^2 + 11x + 6)) y = (x + 3)/(x + 1)`

`\implies` I.F = `e^(intp(x)dx)`  ....(i)

`intp(x)dx = int((2x^2 + 11x + 13)dx)/((x + 1)(x + 2)(x + 3))`

Using partial fraction

`(2x^2 + 11x + 13)/((x + 1)(x + 2)(x + 3)) = P/(x + 1) + Q/(x + 2) + R/(x + 3)`

P = `4/2` = 2

Q = 1

R = –1

∵ `intP(x)dx` = p ln (x + 1)  + Q ln (x + 2) + R ln (x + 3)

= `ln (((x + 1)^2 (x + 2))/(x + 3))`

From (i),

I.F. = `e^(intp(x)dx) = ((x + 1)^2(x + 2))/((x + 3))`

Solution y (IF) = `int Q.(IF)dx`

`y(((x + 1)^2(x + 2))/(x + 3)) = int(\cancel(x + 3)/\cancel(x + 1)) ((x + 1)^2(x + 2))/(\cancel(x + 3))dx`

`y(((x + 1)^2(x + 2))/(x + 3)) = x^3/3 + (3x^2)/2 + 2x + C`

Satisfy (0, 1) then C = `2/3`

Now put x = 1 `\implies` y(1) = `3/2`