`y(x^(2)y+e^(x))dx-e^(x)dy=0`

Explanation

To solve the differential equation given by

\[ y(x^2y + e^x)dx - e^x dy = 0, \]

we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We can rearrange it as follows:

\[ y(x^2y + e^x)dx = e^x dy. \]

### Step 2: Divide by \(y^2\)
Next, we divide the entire equation by \(y^2\) (assuming \(y \neq 0\)):

\[ \frac{x^2y + e^x}{y}dx = \frac{e^x}{y^2}dy. \]

This simplifies to:

\[ \left(x^2 + \frac{e^x}{y}\right)dx = \frac{e^x}{y^2}dy. \]

### Step 3: Rearranging
Rearranging gives us:

\[ \left(x^2 + \frac{e^x}{y}\right)dx - \frac{e^x}{y^2}dy = 0. \]

### Step 4: Identify the form
We can identify this as a differential equation that can be solved using the method of integrating factors or by recognizing it as a separable equation.

### Step 5: Integrate both sides
We can integrate both sides. We have:

\[ \int \left(x^2 + \frac{e^x}{y}\right)dx = \int \frac{e^x}{y^2}dy. \]

Calculating the left side:

\[ \int x^2 dx = \frac{x^3}{3} + C_1, \]

and for the right side, we treat \(y\) as a constant:

\[ \int \frac{e^x}{y^2} dy = \frac{e^x}{y^2} + C_2. \]

### Step 6: Combine results
Combining the results gives us:

\[ \frac{x^3}{3} + C_1 = \frac{e^x}{y^2} + C_2. \]

### Step 7: Rearranging to find the general solution
We can rearrange this to express \(y\) in terms of \(x\):

\[ y^2\left(\frac{x^3}{3} + C_1 - C_2\right) = e^x. \]

Let \(C = C_1 - C_2\), we have:

\[ y^2 = \frac{3e^x}{x^3 + 3C}. \]

Thus, the general solution of the differential equation is:

\[ y = \sqrt{\frac{3e^x}{x^3 + 3C}}. \]