Solution of the differential equation $\frac{dy}{dx}=\frac{y\sqrt{y^{2}-1}}{x\sqrt{x^{2}-1}},$ with $y(2)=\frac{2}{\sqrt{3}}$ is:

Question

Accepted Answer

To solve the differential equation $\frac{dy}{dx} = \frac{y\sqrt{y^{2}-1}}{x\sqrt{x^{2}-1}}$ with the initial condition $y(2) = \frac{2}{\sqrt{3}}$, we need to find a function $y(x)$ that satisfies both the differential equation and the initial condition.

### Explanation for Option 3:

**Option 3: $y = \sec(\sec^{-1}x - \frac{\pi}{6})$**

1. **Separation of Variables:**
   The given differential equation can be separated as:
   $$
   \frac{dy}{y\sqrt{y^2 - 1}} = \frac{dx}{x\sqrt{x^2 - 1}}
   $$
   Integrating both sides, we get:
   $$
   \int \frac{dy}{y\sqrt{y^2 - 1}} = \int \frac{dx}{x\sqrt{x^2 - 1}}
   $$

2. **Integration:**
   The integral $\int \frac{dy}{y\sqrt{y^2 - 1}}$ is known to be $\sec^{-1}(y) + C_1$, and $\int \frac{dx}{x\sqrt{x^2 - 1}}$ is $\sec^{-1}(x) + C_2$. Thus, we have:
   $$
   \sec^{-1}(y) = \sec^{-1}(x) + C
   $$
   where $C = C_2 - C_1$.

3. **Solving for $y$:**
   Solving for $y$, we get:
   $$
   y = \sec(\sec^{-1}(x) + C)
   $$

4. **Applying Initial Condition:**
   Using the initial condition $y(2) = \frac{2}{\sqrt{3}}$, we substitute:
   $$
   \frac{2}{\sqrt{3}} = \sec(\sec^{-1}(2) + C)
   $$
   Solving for $C$, we find that $C = -\frac{\pi}{6}$.

5. **Final Solution:**
   Substituting back, we get:
   $$
   y = \sec(\sec^{-1}(x) - \frac{\pi}{6})
   $$
   This matches Option 3.

### Explanation for Incorrect Options:

**Option 1: $y = \sec^{-1}\frac{2x}{\sqrt{3}} - \sec^{-1}2x$**
- This form does not satisfy the differential equation upon substitution and does not incorporate the initial condition correctly.

**Option 2: $y = \sec^{-1}(\sec~x + \frac{\pi}{6})$**
- This expression is not valid because $\sec^{-1}$ of a sum is not a standard transformation and does not simplify to satisfy the differential equation.

**Option 4: $y = \sec(\sec^{-1}x + \frac{\pi}{3})$**
- This option does not satisfy the initial condition $y(2) = \frac{2}{\sqrt{3}}$ when substituted back into the equation.

In conclusion, Option 3 is the correct solution as it satisfies both the differential equation and the initial condition.