The equation of the curve passing through (3, 9) which satisfies `dy //dx =x +1//x^2 ` is :

Options

A. `6xy =3x ^(2) - 6x + 29`
B. `6 xy = 3x^2 - 29 +6`
C. `6xy = 3x^2 + 29 -6`
D. none of these

Answer

To find the equation of the curve passing through the point (3, 9) that satisfies the differential equation \( \frac{dy}{dx} = x + \frac{1}{x^2} \), we will follow these steps:

### Step 1: Rewrite the differential equation
We start with the given differential equation:
\[ \frac{dy}{dx} = x + \frac{1}{x^2} \]
We can rewrite \( \frac{1}{x^2} \) as \( x^{-2} \):
\[ \frac{dy}{dx} = x + x^{-2} \]

### Step 2: Integrate both sides
Next, we integrate both sides with respect to \( x \):
\[ y = \int \left( x + x^{-2} \right) dx \]
This can be split into two separate integrals:
\[ y = \int x \, dx + \int x^{-2} \, dx \]

### Step 3: Compute the integrals
Now we compute each integral:
1. The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
2. The integral of \( x^{-2} \) is:
\[ \int x^{-2} \, dx = -\frac{1}{x} \]
Combining these results, we have:
\[ y = \frac{x^2}{2} - \frac{1}{x} + C \]
where \( C \) is the constant of integration.

### Step 4: Use the initial condition to find \( C \)
We know the curve passes through the point (3, 9). Therefore, we substitute \( x = 3 \) and \( y = 9 \) into the equation:
\[ 9 = \frac{3^2}{2} - \frac{1}{3} + C \]
Calculating \( \frac{3^2}{2} \):
\[ \frac{3^2}{2} = \frac{9}{2} \]
So we have:
\[ 9 = \frac{9}{2} - \frac{1}{3} + C \]
To solve for \( C \), we first convert all terms to a common denominator, which is 6:
\[ 9 = \frac{27}{6} - \frac{2}{6} + C \]
This simplifies to:
\[ 9 = \frac{25}{6} + C \]
Subtract \( \frac{25}{6} \) from both sides:
\[ C = 9 - \frac{25}{6} = \frac{54}{6} - \frac{25}{6} = \frac{29}{6} \]

### Step 5: Write the final equation
Now we substitute \( C \) back into the equation for \( y \):
\[ y = \frac{x^2}{2} - \frac{1}{x} + \frac{29}{6} \]

### Step 6: Rearranging the equation
To express this equation in a more standard form, we can multiply through by 6 to eliminate the fraction:
\[ 6y = 3x^2 - 6 \cdot \frac{1}{x} + 29 \]
This can be rewritten as:
\[ 6y = 3x^2 + 29 - \frac{6}{x} \]

### Final Equation
Thus, the equation of the curve is:
\[ 6y = 3x^2 + 29 - \frac{6}{x} \]