The solution of `(dy)/(dx)=((y)/(x))^(1//3)`, is

Options

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

Answer

To solve the differential equation

\[ \frac{dy}{dx} = \left(\frac{y}{x}\right)^{\frac{1}{3}}, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We can rewrite the equation as:

\[ \frac{dy}{dx} = \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}. \]

### Step 2: Separating Variables
Next, we will separate the variables \(y\) and \(x\):

\[ \frac{dy}{y^{\frac{1}{3}}} = \frac{dx}{x^{\frac{1}{3}}}. \]

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

\[ \int \frac{dy}{y^{\frac{1}{3}}} = \int \frac{dx}{x^{\frac{1}{3}}}. \]

The left side integrates to:

\[ \int y^{-\frac{1}{3}} dy = \frac{y^{\frac{2}{3}}}{\frac{2}{3}} = \frac{3}{2} y^{\frac{2}{3}} + C_1, \]

and the right side integrates to:

\[ \int x^{-\frac{1}{3}} dx = \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = \frac{3}{2} x^{\frac{2}{3}} + C_2. \]

### Step 4: Setting the Integrals Equal
Setting the two integrals equal gives us:

\[ \frac{3}{2} y^{\frac{2}{3}} = \frac{3}{2} x^{\frac{2}{3}} + C, \]

where \(C = C_2 - C_1\).

### Step 5: Simplifying the Equation
We can simplify this equation by multiplying through by \(\frac{2}{3}\):

\[ y^{\frac{2}{3}} = x^{\frac{2}{3}} + C'. \]

### Step 6: Rearranging the Final Equation
Rearranging gives us:

\[ x^{\frac{2}{3}} - y^{\frac{2}{3}} = -C'. \]

Letting \(C = -C'\), we can write the final solution as:

\[ x^{\frac{2}{3}} - y^{\frac{2}{3}} = C. \]

### Final Answer
Thus, the solution of the differential equation is:

\[ x^{\frac{2}{3}} - y^{\frac{2}{3}} = C, \]

where \(C\) is a constant.

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