Find the general solution of `(dy)/(dx)=sqrt(4-y^(2))(-2 lt y lt 2)`.

Answer

To find the general solution of the differential equation

\[ \frac{dy}{dx} = \sqrt{4 - y^2} \]

where \(-2 < y < 2\), we will follow these steps:

### Step 1: Rearranging the Equation
We start by separating the variables. We can rearrange the equation as follows:

\[ \frac{dy}{\sqrt{4 - y^2}} = dx \]

### Step 2: Integrating Both Sides
Next, we will integrate both sides. The left side requires the integral of \(\frac{dy}{\sqrt{4 - y^2}}\):

\[ \int \frac{dy}{\sqrt{4 - y^2}} = \int dx \]

### Step 3: Using the Integral Formula
The integral \(\int \frac{dy}{\sqrt{a^2 - y^2}}\) is known to be \(\sin^{-1}\left(\frac{y}{a}\right) + C\). Here, \(a = 2\):

\[ \int \frac{dy}{\sqrt{4 - y^2}} = \sin^{-1}\left(\frac{y}{2}\right) + C \]

Thus, we have:

\[ \sin^{-1}\left(\frac{y}{2}\right) = x + C \]

### Step 4: Solving for y
To express \(y\) in terms of \(x\), we take the sine of both sides:

\[ \frac{y}{2} = \sin(x + C) \]

Multiplying through by 2 gives:

\[ y = 2 \sin(x + C) \]

### Step 5: General Solution
The general solution of the given differential equation is:

\[ y = 2 \sin(x + C) \]