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

Options

A. `y=e^(x-y)-x^(2)e^(-y)+C`
B. `e^(y)-e^(x)=(x^(3))/(3)+C`
C. `e^(x)+e^(y)=(x^(3))/(3)+C`
D. `e^(x)-e^(y)=(x^(3))/(3)+C`

Answer

B

Explanation

Given that, `(dy)/(dx)=e^(x-y)+x^(2)e^(-y)`
`Rightarrow (dy)/(dx)=e^(x)e^(-y)+x^(2)e^(-y)`
`Rightarrow (dy)/(dx)=(ex^(2)+x^(2))/(e^(y))`
On integrating both sides, we get
`inte^(y) dy=int(e^(x)+x^(2))dx`
`Rightarrow e^(y)=e^(x)+(x^(3))/(3)+C``Rightarrow e^(y)-e^(x)=(x^(3))/(3)+C`