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Home  >>  CBSE XII  >>  Math  >>  Differential Equations
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The solution of the differential equation $\large\frac{dy}{dx}$$=e^{x-y}+x^2e^{-y}$ is

:\[(A)\;y=e^{x-y}-x^2e^{-y}+c \quad (B)\;e^y-e^x=\frac{x^3}{3}+c \quad(C)\;e^x+e^y=\frac{x^3}{3}+c \quad (D)\;e^x-e^y=\frac{x^3}{3}+c\]

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1 Answer

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Toolbox:
  • If a linear differential equation is of the form $\large\frac{dy}{dx}$$=f(x),$ then it can be solved by seperating the variables.
  • $\int e^xdx=e^x+c$
Given $\large\frac{dy}{dx}$$=e^{x-y}+x^2e^{-y}$
This can be written as
$\large\frac{dy}{dx}=\Large\frac{e^x}{e^y}+\frac{x^2}{e^y}$
$\large\frac{dy}{dx}=\Large\frac{e^x+x^2}{e^y}$
On seperating the variables we get
$e^y.dy=(x^2+e^x)dx$
On integrating we get,
$\int e^y dy=\int x^2 dx+\int e^x dx$
$e^y=\large\frac{x^3}{3}$$+e^x+c$
$e^y-e^x=\large\frac{x^3}{3}+c$
Hence the correct option is $B$
answered May 21, 2013 by meena.p
 
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