The general solution of the differential equation `y(x^(2)y+e^(x))dx-(e^x)dy=0`, is

Options

A. `x^(3)y-3e^(x)=Cy`
B. `x^(3)y+3e^(x)=3Cy`
C. `y^(3)x-3e^(y)=Cx`
D. `y^(3)x+3e^(y)=Cx`

Explanation

To solve the differential equation \( y(x^2 y + e^x)dx - e^x dy = 0 \), we will follow these steps:

### Step 1: Rearranging the equation
We start with the given equation:
\[ y(x^2 y + e^x)dx - e^x dy = 0 \]
We can rearrange this to isolate \( dy \):
\[ y(x^2 y + e^x)dx = e^x dy \]
Dividing both sides by \( e^x \):
\[ \frac{y(x^2 y + e^x)}{e^x}dx = dy \]

### Step 2: Expressing in a more manageable form
We can rewrite the equation as:
\[ \frac{y}{e^x}(x^2 y + e^x)dx = dy \]
This can be simplified to:
\[ \left( \frac{y}{e^x} x^2 y + y \right)dx = dy \]

### Step 3: Separating variables
Now, we will separate the variables:
\[ \frac{y}{e^x} x^2 y dx - dy = 0 \]
This can be rearranged to:
\[ \frac{y}{e^x} x^2 y dx = dy \]

### Step 4: Dividing by \( y^2 \)
Next, we divide both sides by \( y^2 \):
\[ \frac{x^2}{e^x} dx = \frac{dy}{y^2} \]

### Step 5: Integrating both sides
Now we integrate both sides:
\[ \int \frac{x^2}{e^x} dx = \int \frac{dy}{y^2} \]
The right side integrates to:
\[ -\frac{1}{y} + C \]
For the left side, we will use integration by parts or a known result.

### Step 6: Solving the left integral
The integral \( \int \frac{x^2}{e^x} dx \) can be computed using integration by parts twice or using a table of integrals. The result is:
\[ -\frac{x^2}{e^x} + 2\frac{x}{e^x} - 2\frac{1}{e^x} + C \]

### Step 7: Equating both sides
After integrating, we have:
\[ -\frac{x^2}{e^x} + 2\frac{x}{e^x} - 2\frac{1}{e^x} = -\frac{1}{y} + C \]

### Step 8: Rearranging to find the general solution
Now we can rearrange this to find \( y \):
\[ y = \frac{e^x}{C + \frac{x^2}{e^x} - 2\frac{x}{e^x} + 2} \]

### Final Step: Simplifying the expression
This can be further simplified to express \( y \) explicitly in terms of \( x \) and \( C \).

Thus, the general solution of the differential equation is:
\[ x^3 y + 3e^x = C \]