Answer

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**Hint:**

We start solving the problem by bringing the terms of the same variables along with the differential of the same variable on same side (variable separable). We then apply integral on both sides of the obtained equation and we then make use of the results $ \int{adx}=ax+C $ and $ \int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+C $ to get the curve. We then substitute the point $ \left( 3,9 \right) $ in the obtained equation to get the value of the constant of integration to get the required equation of the curve.

**Complete step by step answer:**

According to the problem, we are asked to find the equation of the curve passing through the point $ \left( 3,9 \right) $ and satisfying the differential equation $ \dfrac{dy}{dx}=x+\dfrac{1}{{{x}^{2}}} $ .

Now, we have $ \dfrac{dy}{dx}=x+\dfrac{1}{{{x}^{2}}} $ .

Let us write the equations of same variable on same side along with the differential of the same variable (Variable separable).

$ \Rightarrow dy=\left( x+\dfrac{1}{{{x}^{2}}} \right)dx $ ---(1).

We know that $ \left( a+b \right)dx=adx+bdx $ . Let us use this result in equation (1).

$ \Rightarrow dy=xdx+\dfrac{1}{{{x}^{2}}}dx $ ---(2).

Let us apply integration on both sides of equation (2).

$ \Rightarrow \int{dy}=\int{xdx}+\int{\dfrac{1}{{{x}^{2}}}dx} $ ---(3).

We know that $ \int{adx}=ax+C $ and $ \int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+C $ . Let us use this results in equation (3).

$ \Rightarrow y=\dfrac{{{x}^{1+1}}}{1+1}+\dfrac{{{x}^{-2+1}}}{-2+1}+C $ .

$ \Rightarrow y=\dfrac{{{x}^{2}}}{2}+\dfrac{{{x}^{-1}}}{\left( -1 \right)}+C $ .

$ \Rightarrow y=\dfrac{{{x}^{2}}}{2}-\dfrac{1}{x}+C $ ---(4).

According to the problem, we are given that this curve passes through the point $ \left( 3,9 \right) $ .

$ \Rightarrow 9=\dfrac{{{3}^{2}}}{2}-\dfrac{1}{3}+C $ .

$ \Rightarrow 9=\dfrac{9}{2}-\dfrac{1}{3}+C $ .

$ \Rightarrow 9-\dfrac{9}{2}+\dfrac{1}{3}=C $ .

$ \Rightarrow C=\dfrac{54-27+2}{6} $ .

$ \Rightarrow C=\dfrac{29}{6} $ ---(5).

Let us substitute equation (5) in equation (4).

$ \Rightarrow y=\dfrac{{{x}^{2}}}{2}-\dfrac{1}{x}+\dfrac{29}{6} $ .

$ \Rightarrow y=\dfrac{3{{x}^{3}}-6+29x}{6x} $ .

$ \Rightarrow 6xy=3{{x}^{3}}+29x-6 $ .

So, we have found the equation of the curve as $ 6xy=3{{x}^{3}}+29x-6 $ .

$ \therefore, $ The correct option for the given problem is (c).

**Note:**

We can see that the given problem contains a huge amount of calculation, so we need to perform each step carefully to avoid confusion. We should not forget to add constant of integration while finding the integration as this will give the required answer. We should not make calculation mistakes while solving this problem. Similarly, we can expect problems to find the equation of the curve if it passes through $ \left( 5,9 \right) $ .

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