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. `6xy = 3x^2 - 29x +6`
C. `6y = 3x^2 + 29 -6x`
D. none of these

Answer

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

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

### Step 2: Integrate Both Sides
Now we will integrate both sides:
\[ \int dy = \int \left(x + \frac{1}{x^2}\right) dx \]
The left-hand side integrates to \( y \). For the right-hand side, we can separate the integrals:
\[ y = \int x \, dx + \int \frac{1}{x^2} \, dx \]
Calculating these integrals:
\[ \int x \, dx = \frac{x^2}{2} \]
\[ \int \frac{1}{x^2} \, dx = -\frac{1}{x} \]
Thus, we have:
\[ y = \frac{x^2}{2} - \frac{1}{x} + C \]
where \( C \) is the constant of integration.

### Step 3: Use the Initial Condition
We know the curve passes through the point (3, 9). We can substitute \( x = 3 \) and \( y = 9 \) into our equation to find \( C \):
\[ 9 = \frac{3^2}{2} - \frac{1}{3} + C \]
Calculating \( \frac{3^2}{2} = \frac{9}{2} \) and substituting:
\[ 9 = \frac{9}{2} - \frac{1}{3} + C \]

### Step 4: Solve for C
To solve for \( C \), we first convert everything to a common denominator. The common denominator for 2 and 3 is 6:
\[ 9 = \frac{27}{6} - \frac{2}{6} + C \]
This simplifies to:
\[ 9 = \frac{27 - 2}{6} + C \]
\[ 9 = \frac{25}{6} + C \]
Now, isolating \( C \):
\[ C = 9 - \frac{25}{6} \]
Converting 9 to sixths:
\[ C = \frac{54}{6} - \frac{25}{6} = \frac{29}{6} \]

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

### Step 6: Rearranging the Equation
To express this in a standard form, we can multiply through by \( 6x \) to eliminate the fraction:
\[ 6xy = 3x^3 - 6 + 29x \]
Rearranging gives us:
\[ 6xy = 3x^3 + 29x - 6 \]

### Final Answer
Thus, the equation of the curve is:
\[ 6xy = 3x^3 + 29x - 6 \]