The general solution of differential equation `(dy)/(dx)=e^((x^(2))/(2))+xy` is

Options

A. `y=Ce^(-x^(2)//2)`
B. `y=Ce^(x^(2)//2)`
C. `y=(x+C)e^(x^(2)//2)`
D. `y=(C-x)e^(x^(2)//2)`

Explanation

Given that, `" "(dy)/(dx)=e^(x^(2)//2)+xy`
`rArr" "(dy)/(dx)-xy=e^(x^(2)//2)`
Here, `" "P=-x, Q=e^(x^(2)//2)`
`therefore" "IF=e^(int-xdx)=e^(-x^(2)//2)`
The general sotution is
`" "y*e^(-x^(2)//2)=inte^(-x^(2)//2)-e^(x^(2)//2)dx+C`
`rArr" "ye^(-x^(2)//2)=int1dx+C`
`rArr" "y*e^(-x^(2)//2)=x+C`
`rArr" "y=xe^(-x^(2)//2)+Ce^(+x^(2)//2)`
`rArr" "y=(x+C)e^(x^(2)//2)`