find the particular solution of the differential equation (x^2/(y^2-2))dy/dx=1/2y satisfying the initial condition y(1)=sqrt(3)

We are given the differential equation:

\[
\frac{x^2}{y^2 - 2} \frac{dy}{dx} = \frac{1}{2y}
\]

with the initial condition \( y(1) = \sqrt{3} \).

### Step 1: Separate the variables
We first rewrite the equation to separate the variables \(x\) and \(y\).

\[
\frac{dy}{dx} = \frac{(y^2 - 2)}{2y x^2}
\]

This is a separable equation, so we can now write:

\[
\frac{2y}{y^2 - 2} \, dy = \frac{1}{x^2} \, dx
\]

### Step 2: Integrate both sides

#### Left side:
We use substitution to solve the left side. Let:
\[
u = y^2 - 2 \quad \Rightarrow \quad du = 2y \, dy
\]
Now the integral becomes:
\[
\int \frac{2y}{y^2 - 2} \, dy = \int \frac{du}{u} = \ln |u| + C_1 = \ln |y^2 - 2| + C_1
\]

#### Right side:
The right side is:
\[
\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C_2
\]

### Step 3: Combine the results
We now combine both integrals:
\[
\ln |y^2 - 2| = -\frac{1}{x} + C
\]
where \( C = C_2 - C_1 \) is the constant of integration.

### Step 4: Solve for \(y\)
Exponentiate both sides to eliminate the logarithm:
\[
|y^2 - 2| = e^{-\frac{1}{x} + C} = e^C e^{-\frac{1}{x}} = K e^{-\frac{1}{x}}
\]
where \( K = e^C \) is a new constant. Thus, we have:
\[
y^2 - 2 = \pm K e^{-\frac{1}{x}}
\]
So,
\[
y^2 = 2 \pm K e^{-\frac{1}{x}}
\]

### Step 5: Apply the initial condition
We are given \( y(1) = \sqrt{3} \). Substituting \( x = 1 \) and \( y = \sqrt{3} \) into the equation:
\[
(\sqrt{3})^2 = 2 \pm K e^{-1}
\]
\[
3 = 2 \pm K e^{-1}
\]
\[
K e^{-1} = 1
\]
\[
K = e
\]

### Step 6: Final solution
Substitute \( K = e \) back into the equation:
\[
y^2 = 2 + e^{1 - \frac{1}{x}}
\]
Therefore, the particular solution is:
\[
y = \sqrt{2 + e^{1 - \frac{1}{x}}}
\]

This is the particular solution that satisfies the initial condition \( y(1) = \sqrt{3} \).

Would you like more details on any step, or do you have any further questions?

Here are 5 follow-up questions to explore the topic further:
1. How would the solution change if the initial condition was \( y(1) = 2 \)?
2. What happens if the differential equation is slightly altered, e.g., changing the constant on the right-hand side?
3. Can you solve the same differential equation using a different method like numerical approximation?
4. How do we determine the sign in front of \( K \) in separable differential equations?
5. How does the solution behave as \( x \to \infty \)?

**Tip:** Always check the conditions at multiple points to verify the correctness of the particular solution.

Solve the separable differential equation (x^2/(y^2-2))dy/dx=1/2y with the initial condition y(1)=sqrt(3). This problem involves key concepts from differential equations, including separation of variables and integration, and is ideal for students in undergraduate calculus or differential equations courses.

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