A) \[x+{{e}^{x+y}}=c\]
B) \[x-{{e}^{x+y}}=c\]
C) \[x+{{e}^{-(x+y)}}=c\]
D) \[x-{{e}^{-(x+y)}}=c\]
E) \[x{{e}^{x+y}}+y=c\]
Correct Answer: C
Solution :
Given differential equation is \[\frac{dy}{dx}+1={{e}^{x+y}}\] ...(i) Put \[x+y=t\] Differentiating w.r.t.\[x,\]we get \[1+\frac{dy}{dx}=\frac{dt}{dx}\] ?..(ii) From Eqs. (i) and (ii) \[\frac{dt}{dx}={{e}^{t}}\] \[\Rightarrow \] \[{{e}^{-t}}dt=dx\] \[\Rightarrow \] \[-{{e}^{-t}}=x+c\] \[\Rightarrow \] \[x+{{e}^{-t}}=c\] \[\Rightarrow \] \[x+{{e}^{-(x+y)}}=c\]
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