A) \[\left( \frac{3x+5}{x+1} \right).{{e}^{{{x}^{2}}}}\]
B) \[\left( \frac{6x+5}{x+1} \right).{{e}^{{{x}^{2}}}}\]
C) \[\left( \frac{6x+5}{{{(x+1)}^{2}}} \right).{{e}^{{{x}^{2}}}}\]
D) \[\left( \frac{5-6x}{(x+1)} \right).{{e}^{{{x}^{2}}}}\]
Correct Answer: B
Solution :
The given equation can be written as \[f'(x)-\frac{2x(x+1)}{x+1}f(x)=\frac{{{e}^{x}}}{{{(x+1)}^{2}}}\] i.e., \[f'(x)-2xf(x)=\frac{{{e}^{x}}}{{{(x+1)}^{2}}}\] ? \[\therefore \] I.F. \[={{e}^{\int_{{}}^{{}}{-2xdx}}}={{e}^{-{{x}^{2}}}}\] \[\therefore \] Solution of equation is \[f(x).{{e}^{-{{x}^{2}}}}=\int_{{}}^{{}}{\frac{dx}{{{(x+1)}^{2}}}}\] \[\Rightarrow \,\,f(x).{{e}^{-{{x}^{2}}}}=-\frac{1}{x+1}+c\] At \[x=0,\,\,f(0)=5\,\Rightarrow \,c=6\] \[\therefore \,\,f(x)=\left( \frac{6x+5}{x+1} \right).{{e}^{{{x}^{2}}}}\]
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