`(dy)/(dx) + (xy)/(1-x^(2))=xsqrt(y)`

Explanation

To solve the differential equation

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

we will follow these steps:

### Step 1: Rearranging the Equation
First, we will rearrange the equation to isolate the terms involving \(y\) on one side:

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

### Step 2: Dividing by \(\sqrt{y}\)
Next, we divide the entire equation by \(\sqrt{y}\):

\[ \frac{1}{\sqrt{y}} \frac{dy}{dx} + \frac{xy}{(1 - x^2)\sqrt{y}} = x \]

This can be rewritten as:

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

### Step 3: Substituting \(v = \sqrt{y}\)
Let \(v = \sqrt{y}\). Then, \(y = v^2\) and \(\frac{dy}{dx} = 2v\frac{dv}{dx}\). Substituting this into the equation gives:

\[ \frac{2v}{v} \frac{dv}{dx} + \frac{xv}{1 - x^2} = x \]

This simplifies to:

\[ 2 \frac{dv}{dx} + \frac{xv}{1 - x^2} = x \]

### Step 4: Rearranging the Equation
Now, we can rearrange this into the standard linear form:

\[ \frac{dv}{dx} + \frac{x}{2(1 - x^2)}v = \frac{x}{2} \]

### Step 5: Identifying \(p\) and \(q\)
Here, we identify \(p\) and \(q\):

- \(p = \frac{x}{2(1 - x^2)}\)
- \(q = \frac{x}{2}\)

### Step 6: Finding the Integrating Factor
The integrating factor \(I\) is given by:

\[ I = e^{\int p \, dx} = e^{\int \frac{x}{2(1 - x^2)} \, dx} \]

To compute this integral, we can use substitution. Let \(u = 1 - x^2\), then \(du = -2x \, dx\), and we have:

\[ \int \frac{x}{2(1 - x^2)} \, dx = -\frac{1}{4} \ln |1 - x^2| + C \]

Thus, the integrating factor becomes:

\[ I = e^{-\frac{1}{4} \ln |1 - x^2|} = |1 - x^2|^{-\frac{1}{4}} \]

### Step 7: Multiplying the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor:

\[ |1 - x^2|^{-\frac{1}{4}} \left( \frac{dv}{dx} + \frac{x}{2(1 - x^2)}v \right) = |1 - x^2|^{-\frac{1}{4}} \cdot \frac{x}{2} \]

### Step 8: Integrating Both Sides
Now we integrate both sides:

\[ \int \left( |1 - x^2|^{-\frac{1}{4}} \frac{dv}{dx} + |1 - x^2|^{-\frac{1}{4}} \frac{x}{2(1 - x^2)} v \right) dx = \int |1 - x^2|^{-\frac{1}{4}} \cdot \frac{x}{2} dx \]

### Step 9: Solving the Integral
The left-hand side can be solved as:

\[ \frac{d}{dx} \left( |1 - x^2|^{-\frac{1}{4}} v \right) \]

The right-hand side will require integration techniques, which will yield a function of \(x\) plus a constant.

### Step 10: Back Substituting for \(y\)
Finally, we substitute back \(v = \sqrt{y}\) to find \(y\) in terms of \(x\).