Equation of the curve whose slope at the point (x,y) is -(x + y) / x and which passes through the points (2, 1) is

Options

A. `2y^(2)+xy=4`
B. `y^(2)+xy=3`
C. `x^(2)+2xy=8`
D. None of these

Answer

To find the equation of the curve whose slope at the point (x, y) is given by the expression \(-\frac{x + y}{x}\) and which passes through the point (2, 1), we can follow these steps:

### Step 1: Set up the differential equation

The slope of the curve at any point (x, y) is given by the derivative \(\frac{dy}{dx}\). Therefore, we can write the differential equation as:

\[ \frac{dy}{dx} = -\frac{x + y}{x} \]

### Step 2: Simplify the equation

We can simplify the right-hand side:

\[ \frac{dy}{dx} = -\frac{x}{x} - \frac{y}{x} = -1 - \frac{y}{x} \]

Thus, we have:

\[ \frac{dy}{dx} = -1 - \frac{y}{x} \]

### Step 3: Rearranging the equation

Rearranging gives us:

\[ \frac{dy}{dx} + \frac{y}{x} = -1 \]

This is a linear first-order differential equation in standard form.

### Step 4: Find the integrating factor

To solve this differential equation, we need to find the integrating factor, \( \mu(x) \):

\[ \mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln |x|} = |x| \]

Since we are considering positive values of \(x\), we can take \( \mu(x) = x \).

### Step 5: Multiply through by the integrating factor

Multiplying the entire equation by \(x\):

\[ x \frac{dy}{dx} + y = -x \]

### Step 6: Recognize the left-hand side as a derivative

The left-hand side can be recognized as the derivative of a product:

\[ \frac{d}{dx}(xy) = -x \]

### Step 7: Integrate both sides

Now, we integrate both sides with respect to \(x\):

\[ \int \frac{d}{dx}(xy) \, dx = \int -x \, dx \]

This gives us:

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

### Step 8: Solve for \(y\)

Now, we can solve for \(y\):

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

### Step 9: Use the initial condition

We know the curve passes through the point (2, 1). We can use this to find \(C\):

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

This simplifies to:

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

Adding 1 to both sides:

\[ 2 = \frac{C}{2} \]

Multiplying both sides by 2 gives:

\[ C = 4 \]

### Step 10: Write the final equation

Substituting \(C\) back into the equation for \(y\):

\[ y = -\frac{x}{2} + \frac{4}{x} \]

Thus, the equation of the curve is:

\[ y = -\frac{x}{2} + \frac{4}{x} \]

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