The solution of differential equation `(1-xy + x^(2) y^(2))dx = x^(2) dy` is

Options

A. tan xy = log |cx|
B. tan (y/x) = tan log |cx|
C. xy = tan log |cx|
D. None of these

Explanation

To solve the differential equation \((1 - xy + x^2y^2)dx = x^2 dy\), we will follow a systematic approach.

### Step 1: Rearranging the Equation
First, we can rearrange the given equation to isolate the differentials:
\[ (1 - xy + x^2y^2)dx - x^2 dy = 0 \]

### Step 2: Introducing a Substitution
Let's introduce a substitution to simplify the equation. Let \(v = xy\). Then, we can express \(y\) in terms of \(v\) and \(x\):
\[ y = \frac{v}{x} \]

### Step 3: Differentiating the Substitution
Now, we differentiate \(v = xy\) with respect to \(x\):
\[ \frac{dv}{dx} = y + x\frac{dy}{dx} \]
Substituting \(y = \frac{v}{x}\) into the equation gives:
\[ \frac{dv}{dx} = \frac{v}{x} + x\frac{dy}{dx} \]

### Step 4: Substituting Back into the Equation
Substituting \(y\) and \(\frac{dy}{dx}\) into the original equation:
\[ (1 - v + v^2)dx = x^2\left(\frac{dv}{dx} - \frac{v}{x^2}\right)dx \]
This simplifies to:
\[ (1 - v + v^2)dx = x^2\frac{dv}{dx}dx - vdx \]

### Step 5: Rearranging Terms
Rearranging the terms gives:
\[ (1 - v + v^2 + v)dx = x^2\frac{dv}{dx}dx \]
This simplifies to:
\[ (1 + v^2)dx = x^2\frac{dv}{dx}dx \]

### Step 6: Separating Variables
Now, we can separate the variables:
\[ \frac{dx}{x^2} = \frac{dv}{1 + v^2} \]

### Step 7: Integrating Both Sides
Integrating both sides:
\[ \int \frac{dx}{x^2} = \int \frac{dv}{1 + v^2} \]
The left side integrates to:
\[ -\frac{1}{x} + C_1 \]
The right side integrates to:
\[ \tan^{-1}(v) + C_2 \]

### Step 8: Combining Results
Combining the results gives:
\[ -\frac{1}{x} = \tan^{-1}(v) + C \]
where \(C = C_2 - C_1\).

### Step 9: Substituting Back for \(v\)
Substituting back \(v = xy\):
\[ -\frac{1}{x} = \tan^{-1}(xy) + C \]

### Final Solution
Thus, the solution of the differential equation is:
\[ \tan^{-1}(xy) = -\frac{1}{x} - C \]