`(1)/(x)dy-(1)/(y^(3))dx=0`

Options

A. `y^(2)-4x^(4)=c`
B. `y^(2)+4x^(4)=c`
C. `y^(4)-x^(2)=c`
D. `2x^(2)-y^(4)=c`

Answer

To solve the differential equation \(\frac{1}{x} dy - \frac{1}{y^3} dx = 0\), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate the differentials:

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

### Step 2: Cross Multiplying
Next, we can cross-multiply to get:

\[ y^3 dy = x dx \]

### Step 3: Integrating Both Sides
Now we will integrate both sides. The left side integrates \(y^3 dy\) and the right side integrates \(x dx\):

\[ \int y^3 dy = \int x dx \]

Calculating the integrals, we have:

\[ \frac{y^4}{4} = \frac{x^2}{2} + C \]

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

### Step 4: Simplifying the Equation
To simplify, we can multiply the entire equation by 4 to eliminate the fraction:

\[ y^4 = 2x^2 + 4C \]

### Step 5: Defining the Constant
We can let \(4C\) be a new constant, say \(A\):

\[ y^4 = 2x^2 + A \]

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

\[ y^4 = 2x^2 + A \]