The solution of `(dy)/(dx) + (x^(2))/(y^(2)) = 0` is

Options

A. `x^(3) + y^(2) = 7`
B. `x^(2) + y^(2) = c`
C. `x^(3) + y^(3) = c`
D. `x + y = c`

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{x^2}{y^2} = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate \(\frac{dy}{dx}\):

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

**Hint:** Move the term involving \(y\) to the right side to isolate \(\frac{dy}{dx}\).

### Step 2: Cross Multiplying
Next, we cross-multiply to separate the variables:

\[ y^2 \, dy = -x^2 \, dx. \]

**Hint:** This step involves multiplying both sides by \(y^2\) and \(dx\) to separate the variables.

### Step 3: Integrating Both Sides
Now, we integrate both sides:

\[ \int y^2 \, dy = \int -x^2 \, dx. \]

Calculating the integrals, we have:

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

where \(C\) is the constant of integration.

**Hint:** Remember the integral of \(y^n\) is \(\frac{y^{n+1}}{n+1}\).

### Step 4: Simplifying the Equation
To simplify, we multiply the entire equation by 3:

\[ y^3 = -x^3 + 3C. \]

**Hint:** Multiplying by a constant helps eliminate fractions and simplifies the expression.

### Step 5: Rearranging to Standard Form
Rearranging the equation gives us:

\[ x^3 + y^3 = 3C. \]

Letting \(C' = 3C\), we can express the solution as:

\[ x^3 + y^3 = C', \]

where \(C'\) is a new constant.

**Hint:** You can redefine the constant to simplify the final expression.

### Final Solution
Thus, the solution to the differential equation is:

\[ x^3 + y^3 = C. \]

### Conclusion
The correct option from the given choices is \(x^3 + y^3 = C\).

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