Question

The solution of differential equation dy/dx = e^{x - y} + x²e^-y is
1. e^y = e^x + x³/3 + c
2. d^y - e^x = c
3. x - e^y = c
4. e^y + e^x + x³/3 + y = 0

Answer 1

Correct Answer - Option 1 : e^y = e^x + x³/3 + c

Given: \(\frac{{dy}}{{dx}} = {e^{x - y}} + {x^2}{e^{ - y}}\)

\(\frac{{dy}}{{dx}} = {e^x}.{e^{ - y}} + {x^2}{e^{ - y}}\) 

\(\frac{{dy}}{{dx}} = {e^{ - y}}\left( {{e^x} + {x^2}} \right)\) 

\(\frac{{dy}}{{dx}} = \frac{{{e^x} + {x^2}}}{{ey}}\) 

dy.ey = (e^x + x²) dx → this is variable separable form taking integration on both side.

\(\smallint {e^y}dy = \smallint {e^x}dx + \smallint {x^2}dx\) 

\({e^y} = {e^x} + \frac{{{x^3}}}{3} + c\) 

∵ \(\smallint {x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}\)  

∵ ∫ e^xdx = e^x