A) \[\log \,(5+10y)=x+\frac{{{x}^{2}}}{2}+c\]
B) \[\frac{1}{10}\log \,(5+10y)=x+\frac{{{x}^{2}}}{2}+c\]
C) \[\frac{1}{5}\log \,(5+10y)=x+\frac{{{x}^{2}}}{2}+c\]
D) \[\frac{1}{10}\log \,(5+10y)=x+{{x}^{2}}+c\]
Correct Answer: B
Solution :
Given, differential equation is \[\frac{dy}{dx}=5+5x+10y+10xy\] \[\Rightarrow \] \[\frac{dy}{dx}=5(1+x)\,+10y\,(1+x)\] \[=(1+x)\,(5+10y)\] \[\Rightarrow \] \[\frac{dy}{5+10y}=(1+x)\,dx\] On integrating \[\Rightarrow \] \[\int{\frac{dy}{5+10y}\,=\int{(1+x)\,dx}}\] \[\Rightarrow \] \[\frac{1}{10}\,\log \,(5+10y)=x+\frac{{{x}^{2}}}{2}+c\]
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