Solve the differential equation `(x^(2)-yx^(2))dy +(y^(2)+x^(2)y^(2))dx =0,` given that y = 1 when x = 1.

Options

A. `log|2y|+(1)/(y)+(1)/(x)-x=1`
B. `log|3y|+(1)/(y)+(1)/(x)-x=1`
C. `log|x|+(1)/(y)+(1)/(x)-x=1`
D. `log|y|+(1)/(y)+(1)/(x)-x=1`

Explanation

To solve the differential equation

\[ (x^{2} - yx^{2})dy + (y^{2} + x^{2}y^{2})dx = 0, \]

given that \( y = 1 \) when \( x = 1 \), we will follow these steps:

### Step 1: Rewrite the Equation
We can rearrange the equation to isolate the differentials:

\[ (x^{2} - yx^{2})dy = -(y^{2} + x^{2}y^{2})dx. \]

### Step 2: Factor Out Common Terms
Next, we can factor out \( x^{2} \) from the left-hand side and \( y^{2} \) from the right-hand side:

\[ (1 - y) x^{2} dy = -(y^{2}(1 + x^{2}))dx. \]

### Step 3: Separate Variables
Now, we can separate the variables \( y \) and \( x \):

\[ \frac{1 - y}{y^{2}} dy = -\frac{1 + x^{2}}{x^{2}} dx. \]

### Step 4: Integrate Both Sides
Now we integrate both sides:

\[ \int \frac{1 - y}{y^{2}} dy = \int -\frac{1 + x^{2}}{x^{2}} dx. \]

The left-hand side can be split into two integrals:

\[ \int \left( \frac{1}{y^{2}} - \frac{1}{y} \right) dy = \int -\left( \frac{1}{x^{2}} + 1 \right) dx. \]

### Step 5: Solve the Integrals
Calculating the integrals:

- Left-hand side:
\[ \int \frac{1}{y^{2}} dy - \int \frac{1}{y} dy = -\frac{1}{y} - \log |y| + C_1. \]

- Right-hand side:
\[ -\int \frac{1}{x^{2}} dx - \int 1 dx = \frac{1}{x} - x + C_2. \]

### Step 6: Combine the Results
Combining the results, we have:

\[ -\frac{1}{y} - \log |y| = \frac{1}{x} - x + C, \]

where \( C = C_2 - C_1 \).

### Step 7: Find the Particular Solution
Now, we use the initial condition \( y = 1 \) when \( x = 1 \):

Substituting \( x = 1 \) and \( y = 1 \):

\[ -\frac{1}{1} - \log(1) = \frac{1}{1} - 1 + C. \]

This simplifies to:

\[ -1 - 0 = 1 - 1 + C \implies -1 = C. \]

### Step 8: Write the Final Solution
Substituting \( C \) back into the equation gives us:

\[ -\frac{1}{y} - \log |y| = \frac{1}{x} - x - 1. \]

This is the particular solution to the differential equation.