A function `y=f(x)` satisfies `(x+1)f^(prime)(x)-2(x^2+x)f(x)=(e^x^2)/((x+1)),AAx > -1.` If `f(0)=5,` then `f(x)` is

Options

A. `((3x+5)/(x+1))e^(x^(2))`
B. `((6x+5)/(x+1))e^(x^(2))`
C. `((6x+5)/(x+1)^(2))e^(x^(2))`
D. `((5-6x)/(x+1))e^(x^(2))`

Answer

B

Explanation

`f^(')(x) - (2x(x+1))/(x+1)f(x)=e^(x^(2))/(x+1)^(2)`
I.F. `=e^(int-2xdx)=e^(-x^(2))`
Thus, the solution is `f(x)e^(-x^(2))=int(dx)/(x+1)^(2)+C`
or `f(x)e^(-x^(2))=int(dx)/(x+1)^(2)+C`
Given, f(0) =5. Thus, C=6
`therefore f(x) = ((6x+5)/(x+1))e^(x^(2))`