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

Options

A. `17 x^(2) y - 6x^(2) + 3x^(5) -2 =0`
B. `6x^(2) y + 17x^(2) + 2x^(5) - 3 =0`
C. `6xy - 2x^(2) + 17 x^(5) + 3 = 0`
D. `17 x^(2) y + 6xy - 3x^(5) + 5 = 0`

Answer

To solve the problem, we need to find the equation of the curve that passes through the point \((-1, -2)\) and satisfies the differential equation:

\[ \frac{dy}{dx} = -x^2 - \frac{1}{x^3} \]

### Step 1: Rewrite the differential equation
We can rewrite the differential equation as:

\[ \frac{dy}{dx} = -x^2 - x^{-3} \]

### Step 2: Separate the variables
We separate the variables \(y\) and \(x\):

\[ dy = \left(-x^2 - x^{-3}\right)dx \]

### Step 3: Integrate both sides
Now we integrate both sides:

\[ \int dy = \int \left(-x^2 - x^{-3}\right)dx \]

The left side integrates to \(y\). For the right side, we integrate term by term:

\[ y = -\frac{x^3}{3} + \frac{x^{-2}}{2} + C \]

This simplifies to:

\[ y = -\frac{x^3}{3} + \frac{1}{2x^2} + C \]

### Step 4: Apply the initial condition
We know the curve passes through the point \((-1, -2)\). We substitute \(x = -1\) and \(y = -2\) into the equation to find \(C\):

\[ -2 = -\frac{(-1)^3}{3} + \frac{1}{2(-1)^2} + C \]

Calculating the right side:

\[ -2 = \frac{1}{3} + \frac{1}{2} + C \]

Finding a common denominator (6):

\[ -2 = \frac{2}{6} + \frac{3}{6} + C \]

This simplifies to:

\[ -2 = \frac{5}{6} + C \]

### Step 5: Solve for \(C\)
Rearranging gives:

\[ C = -2 - \frac{5}{6} = -\frac{12}{6} - \frac{5}{6} = -\frac{17}{6} \]

### Step 6: Substitute \(C\) back into the equation
Now we substitute \(C\) back into our equation for \(y\):

\[ y = -\frac{x^3}{3} + \frac{1}{2x^2} - \frac{17}{6} \]

### Step 7: Rearranging the equation
To express it in a standard form, we can multiply through by \(6x^2\) to eliminate the fractions:

\[ 6x^2y = -2x^5 + 3 - 17x^2 \]

Rearranging gives:

\[ 2x^5 + 6x^2y + 17x^2 - 3 = 0 \]

This is the equation of the curve.

### Final Answer
The equation of the curve is:

\[ 2x^5 + 6x^2y + 17x^2 - 3 = 0 \]

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