The differential equation `(dy)/(dx)=(2x)/(3y)` represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity. (a)`sqrt(3/5)` (b) `sqrt(5/3)` (c)`sqrt(2/5)` (d) `sqrt(5/2)`

Options

A. `sqrt(3//5)`
B. `sqrt(5//3)`
C. `sqrt(2//5)`
D. `sqrt(5//2)`

Explanation

To solve the differential equation \(\frac{dy}{dx} = \frac{2x}{3y}\) and find the eccentricity of the family of hyperbolas it represents, we can follow these steps:

### Step 1: Separate the Variables
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{2x}{3y} \]
We can separate the variables by multiplying both sides by \(3y\) and \(dx\):
\[ 3y \, dy = 2x \, dx \]

### Step 2: Integrate Both Sides
Now we integrate both sides:
\[ \int 3y \, dy = \int 2x \, dx \]
The left side integrates to:
\[ \frac{3y^2}{2} + C_1 \]
And the right side integrates to:
\[ x^2 + C_2 \]
Combining these, we have:
\[ \frac{3y^2}{2} = x^2 + C \]
where \(C = C_2 - C_1\).

### Step 3: Rearranging the Equation
Rearranging the equation gives us:
\[ 3y^2 = 2x^2 + 2C \]
Dividing through by \(2C\) (assuming \(C \neq 0\)):
\[ \frac{y^2}{\frac{2C}{3}} - \frac{x^2}{C} = 1 \]
This is in the standard form of a hyperbola:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]
where \(a^2 = \frac{2C}{3}\) and \(b^2 = C\).

### Step 4: Finding the Eccentricity
The eccentricity \(e\) of a hyperbola is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ e = \sqrt{1 + \frac{C}{\frac{2C}{3}}} \]
This simplifies to:
\[ e = \sqrt{1 + \frac{3}{2}} = \sqrt{\frac{5}{2}} \]

### Step 5: Consider the Case for \(C < 0\)
If \(C < 0\), we can repeat the steps:
The equation becomes:
\[ \frac{y^2}{\frac{2(-C)}{3}} - \frac{x^2}{-C} = 1 \]
This gives:
\[ e = \sqrt{1 + \frac{-C}{\frac{2(-C)}{3}}} = \sqrt{1 + \frac{3}{2}} = \sqrt{\frac{5}{3}} \]

### Conclusion
Thus, the eccentricities of the hyperbolas represented by the differential equation are:
- For \(C > 0\), \(e = \sqrt{\frac{5}{2}}\)
- For \(C < 0\), \(e = \sqrt{\frac{5}{3}}\)

The correct options are:
(b) \(\sqrt{\frac{5}{3}}\) and (d) \(\sqrt{\frac{5}{2}}\).