Solve the following differential equations
`ye^(y)dx= (y^3+2xe^(y))dy`.

Answer

To solve the differential equation \( y e^{y} dx = (y^3 + 2x e^{y}) dy \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ y e^{y} dx = (y^3 + 2x e^{y}) dy \]
We can rearrange this to express \( dx \) in terms of \( dy \):
\[ dx = \frac{(y^3 + 2x e^{y})}{y e^{y}} dy \]

### Step 2: Express \( \frac{dx}{dy} \)
Now, we can express \( \frac{dx}{dy} \):
\[ \frac{dx}{dy} = \frac{y^3}{y e^{y}} + \frac{2x e^{y}}{y e^{y}} = \frac{y^2}{e^{y}} + \frac{2x}{y} \]

### Step 3: Rearrange into standard linear form
We can rearrange this equation to match the standard form of a linear differential equation:
\[ \frac{dx}{dy} - \frac{2}{y} x = \frac{y^2}{e^{y}} \]
Here, we identify \( p(y) = -\frac{2}{y} \) and \( q(y) = \frac{y^2}{e^{y}} \).

### Step 4: Find the integrating factor
The integrating factor \( \mu(y) \) is given by:
\[ \mu(y) = e^{\int p(y) dy} = e^{\int -\frac{2}{y} dy} = e^{-2 \ln |y|} = |y|^{-2} \]
Since \( y \) is positive in our context, we can write:
\[ \mu(y) = \frac{1}{y^2} \]

### Step 5: Multiply through by the integrating factor
Multiply the entire differential equation by the integrating factor:
\[ \frac{1}{y^2} \frac{dx}{dy} - \frac{2}{y^3} x = \frac{y^2}{y^2 e^{y}} = \frac{1}{e^{y}} \]

### Step 6: Recognize the left side as a derivative
The left side can be recognized as the derivative of a product:
\[ \frac{d}{dy} \left( \frac{x}{y^2} \right) = \frac{1}{e^{y}} \]

### Step 7: Integrate both sides
Integrate both sides with respect to \( y \):
\[ \int \frac{d}{dy} \left( \frac{x}{y^2} \right) dy = \int \frac{1}{e^{y}} dy \]
This gives:
\[ \frac{x}{y^2} = -e^{-y} + C \]
where \( C \) is the constant of integration.

### Step 8: Solve for \( x \)
Now, we solve for \( x \):
\[ x = y^2 (-e^{-y} + C) = -y^2 e^{-y} + C y^2 \]

### Final Solution
Thus, the solution to the differential equation is:
\[ x = -y^2 e^{-y} + C y^2 \]