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

Options

A. `x ^(3) + y ^(3) =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 can follow these steps:

### Step 1: Rearrange the equation
We start by rearranging the equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{x^2}{y^2} \]

**Hint:** Rearranging the equation helps to isolate the derivative on one side.

### Step 2: Separate the variables
Next, we separate the variables \(y\) and \(x\):
\[ y^2 \, dy = -x^2 \, dx \]

**Hint:** To separate variables, multiply both sides by the appropriate terms to get all \(y\) terms on one side and all \(x\) terms on the other.

### Step 3: Integrate both sides
Now, we integrate both sides:
\[ \int y^2 \, dy = -\int x^2 \, dx \]

Calculating the integrals, we get:
\[ \frac{y^3}{3} = -\frac{x^3}{3} + C \]

**Hint:** Remember to add the constant of integration \(C\) after integrating.

### Step 4: Simplify the equation
Multiplying through by 3 to eliminate the fractions gives:
\[ y^3 + x^3 = 3C \]

**Hint:** Simplifying the equation can help in identifying the final form of the solution.

### Step 5: Express the solution
We can express the solution in a more standard form:
\[ x^3 + y^3 = k \]
where \(k = 3C\) is a constant.

**Hint:** The final solution can be expressed in terms of a new constant for clarity.

### Final Solution
The solution of the differential equation \(\frac{dy}{dx} + \frac{x^2}{y^2} = 0\) is:
\[ x^3 + y^3 = C \]
where \(C\) is a constant.

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