The general solution of the differential equation `(dy)/(dx)=x^2/y^2` is

Options

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

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{x^2}{y^2}\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{x^2}{y^2} \]

### Step 2: Cross-multiply
To separate the variables, we can cross-multiply:
\[ y^2 \, dy = x^2 \, dx \]

### Step 3: Integrate both sides
Now, we will integrate both sides. The left side integrates with respect to \(y\) and the right side integrates with respect to \(x\):
\[ \int y^2 \, dy = \int x^2 \, dx \]

### Step 4: Perform the integration
Calculating the integrals:
\[ \frac{y^3}{3} = \frac{x^3}{3} + C \]
where \(C\) is the constant of integration.

### Step 5: Rearranging the equation
To express the solution in a standard form, we can multiply through by 3:
\[ y^3 = x^3 + 3C \]

### Step 6: Final form of the general solution
We can rewrite the equation as:
\[ x^3 - y^3 = -3C \]
Letting \(C' = -3C\) (where \(C'\) is also a constant), we can express the general solution as:
\[ x^3 - y^3 = C' \]

Thus, the general solution of the differential equation \(\frac{dy}{dx} = \frac{x^2}{y^2}\) is:
\[ x^3 - y^3 = C \]

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