A) \[(1+{{x}^{2}})y={{x}^{3}}\]
B) \[3(1+{{x}^{2}})y=2{{x}^{3}}\]
C) \[1(1+{{x}^{2}})y=3{{x}^{3}}\]
D) \[3(1+{{x}^{2}})y=4{{x}^{3}}\]
Correct Answer: D
Solution :
Given differential equation is \[(1+{{x}^{2}})\frac{dy}{dx}+2xy=4{{x}^{2}}\] \[\Rightarrow \]\[\frac{dy}{dx}+\left( \frac{2x}{1+{{x}^{2}}} \right)y=\frac{4{{x}^{2}}}{1+{{x}^{2}}}\] This is linear diff. equation \[I.F={{e}^{\int_{{}}^{{}}{\frac{2x}{1+{{x}^{2}}}dx}}}={{e}^{\log (1+{{x}^{2}})}}=1+{{x}^{2}}\] \[\therefore \]Solution is \[y(1+{{x}^{2}})=\int_{{}}^{{}}{\frac{4{{x}^{2}}}{1+{{x}^{2}}}\times 1+{{x}^{2}}+C}\] \[\Rightarrow \]\[y(1+{{x}^{2}})=\frac{4{{x}^{3}}}{3}+C\] \[\Rightarrow \]Required curve is \[3y(1+{{x}^{2}})=4{{x}^{3}}(\because C=0)\]
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