Solve: `e^xsqrt(1-y^2)dx+y/x dy=0`

Explanation

To solve the differential equation \( e^x \sqrt{1 - y^2} \, dx + \frac{y}{x} \, dy = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to separate the variables \(x\) and \(y\):

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

### Step 2: Separating Variables
Next, we can rewrite the equation in a form that separates the variables:

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

Now, we can multiply both sides by \(x\):

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

### Step 3: Integrating Both Sides
Now we can integrate both sides. We will integrate the left side with respect to \(x\) and the right side with respect to \(y\):

\[ \int x e^x \, dx = -\int y \, dy \]

### Step 4: Solving the Left Integral
To solve the left integral, we can use integration by parts. Let:
- \(u = x\) and \(dv = e^x \, dx\)
- Then, \(du = dx\) and \(v = e^x\)

Using integration by parts:

\[ \int u \, dv = uv - \int v \, du \]

We have:

\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C_1 \]

### Step 5: Solving the Right Integral
Now, we integrate the right side:

\[ -\int y \, dy = -\frac{y^2}{2} + C_2 \]

### Step 6: Equating Both Integrals
Now we equate both sides:

\[ x e^x - e^x = -\frac{y^2}{2} + C \]

where \(C = C_2 - C_1\).

### Step 7: Rearranging the Equation
Rearranging gives us:

\[ x e^x - e^x + \frac{y^2}{2} = C \]

### Step 8: Final Form
This is the implicit solution of the differential equation. We can express it as:

\[ x e^x - e^x + \frac{y^2}{2} = C \]

### Summary
The solution to the differential equation \( e^x \sqrt{1 - y^2} \, dx + \frac{y}{x} \, dy = 0 \) is given by:

\[ x e^x - e^x + \frac{y^2}{2} = C \]