We are given the differential equation

$\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}},$

and an initial condition $y = \frac{\sqrt{3}}{2}$ when $x = \frac{1}{2}$. We are asked to show that

$2y = x \sqrt{k} + \sqrt{1 - x^2},$

where $k$ is a constant to be found.

Step-by-Step Solution

1. Separate the variables: We rewrite the differential equation to separate $x$ and $y$.

$\frac{dy}{\sqrt{1 - y^2}} = \frac{dx}{\sqrt{1 - x^2}}.$

2. Integrate both sides: We integrate both sides with respect to $y$ and $x$:

$\int \frac{dy}{\sqrt{1 - y^2}} = \int \frac{dx}{\sqrt{1 - x^2}}.$

The left-hand side is a standard integral:

$\int \frac{dy}{\sqrt{1 - y^2}} = \arcsin(y),$

and similarly for the right-hand side:

$\int \frac{dx}{\sqrt{1 - x^2}} = \arcsin(x).$

Thus, the equation becomes:

$\arcsin(y) = \arcsin(x) + C,$

where $C$ is the constant of integration.

3. Solve for $y$: Taking the sine of both sides:

$y = \sin(\arcsin(x) + C).$

Using the trigonometric identity for the sine of a sum:

$y = \sin(\arcsin(x)) \cos(C) + \cos(\arcsin(x)) \sin(C).$

Since $\sin(\arcsin(x)) = x$ and $\cos(\arcsin(x)) = \sqrt{1 - x^2}$, this simplifies to:

$y = x \cos(C) + \sqrt{1 - x^2} \sin(C).$

4. Multiply by 2: To match the given form, multiply both sides by 2:

$2y = 2x \cos(C) + 2 \sqrt{1 - x^2} \sin(C).$

Comparing this with the target expression $2y = x \sqrt{k} + \sqrt{1 - x^2}$, we identify:

$\sqrt{k} = 2 \cos(C) \quad \text{and} \quad 1 = 2 \sin(C).$

From $2 \sin(C) = 1$, we find:

$\sin(C) = \frac{1}{2}.$

Thus, $C = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$.

5. Find $k$: Using $\sin(C) = \frac{1}{2}$ and $\cos(C) = \frac{\sqrt{3}}{2}$, we have:

$\sqrt{k} = 2 \cos\left(\frac{\pi}{6}\right) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}.$

Therefore, $k = 3$.

Final Answer:

Thus, the equation becomes:

$2y = x \sqrt{3} + \sqrt{1 - x^2}.$

Let me know if you'd like further clarification on any part of this! Would you also like more details on any of the steps?

Here are 5 related questions to extend your understanding:

1. How can we interpret the constant $C$ in terms of boundary conditions?
2. What other methods can be used to solve this differential equation?
3. How does the chain rule apply in the separation of variables?
4. Can you derive the identity for $\sin(\arcsin(x) + C)$ directly?
5. What are some applications of this type of differential equation in physics?

Tip: When solving differential equations, always check the boundary conditions after integrating to ensure the correct constant of integration.