Select the correct option for the solution of the differential equation \[\dfrac{{dy}}{{dx}} = {e^{x - y}} + {x^2}{e^{^{ - y}}}\].
A. \[{e^y} = {e^x} + \dfrac{{{x^3}}}{3} + c\]
B. \[{e^y} = {e^x} + 2x + c\]
C. \[{e^y} = {e^x} + {x^3} + c\]
D. \[y = {e^x} + c\]

Hint: Use variable separation methods to find the solution of the differential equation. X and y are two variables in this differential equation. Separate the x containing terms and y containing terms. Then integrate it to find the general solution.

Formula used:
\[\begin{array}{l}\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\\\int {{e^x}dx} = {e^x} + c\end{array}\]
Where, c is an arbitrary constant.

Complete step by step solution:
The given differential equation is \[\dfrac{{dy}}{{dx}} = {e^{x - y}} + {x^2}{e^{^{ - y}}}\].
First simplify the equation by taking \[{e^{ - y}}\] common from the right hand side of the equation.
\[\dfrac{{dy}}{{dx}} = {e^{ - y}}\left( {{e^x} + {x^2}} \right)\]
Now multiply both sides of the equation by \[{e^y}dx\] and simplify it.
\[\begin{array}{l}\left( {\dfrac{{dy}}{{dx}}} \right){e^y}dx = {e^{ - y}}\left( {{e^x} + {x^2}} \right){e^y}dx\\{e^y}dy = \left( {{e^x} + {x^2}} \right)dx\end{array}\]
Here integrate the equation.
\[\begin{array}{l}\int {{e^y}dy} = \int {\left( {{e^x} + {x^2}} \right)dx} \\{e^y} = \int {{e^x}dx + \int {{x^2}dx} } \end{array}\]
Simplify as follows to find a general solution.
\[{e^y} = {e^x} + \dfrac{{{x^3}}}{3} + c\]
So, the general solution of the differential equation is \[{e^y} = {e^x} + \dfrac{{{x^3}}}{3} + c\].
Hence, the correct answer is Option A.

Note: The foremost often mistake done here is application of formula. Often, we use \[\int {{e^y}} dx = \ln x + c\] which is wrong. Also, \[{e^{x - y}}\] is not \[{e^x} - {e^y}\]. This mistake happens mostly.