The solution of the differential equation
` ( 3xy +y^(2)) dx + (x^(2) +xy)dy = 0 ` is

Options

A. `x^(2)(2xy+y^(2))=C^(2)`
B. `x^(2)(2xy-y^(2))=C^(2)`
C. `x^(2)(y^(2)-2xy)=C^(2)`
D. None of the above

Explanation

To solve the differential equation \( (3xy + y^2)dx + (x^2 + xy)dy = 0 \), we will follow these steps:

### Step 1: Identify the type of differential equation
The given differential equation is:
\[ (3xy + y^2)dx + (x^2 + xy)dy = 0 \]
We can check if this is a homogeneous equation by observing the degrees of the terms. The degree of each term can be calculated as follows:
- \(3xy\) has degree 2.
- \(y^2\) has degree 2.
- \(x^2\) has degree 2.
- \(xy\) has degree 2.

Since all terms are of degree 2, the equation is homogeneous.

**Hint:** Check the degrees of all terms to confirm if the equation is homogeneous.

### Step 2: Substitute \(y = vx\)
To solve the homogeneous equation, we substitute \(y = vx\), where \(v\) is a function of \(x\). Then, we differentiate \(y\) with respect to \(x\):
\[ dy = vdx + xdv \]

**Hint:** Use the substitution \(y = vx\) to express \(dy\) in terms of \(dx\) and \(dv\).

### Step 3: Substitute into the differential equation
Substituting \(y = vx\) and \(dy = vdx + xdv\) into the original equation gives:
\[ (3x(vx) + (vx)^2)dx + (x^2 + x(vx))(vdx + xdv) = 0 \]
This simplifies to:
\[ (3vx^2 + v^2x^2)dx + (x^2 + vx^2)(vdx + xdv) = 0 \]

### Step 4: Simplify the equation
Factor out \(x^2\) from the terms:
\[ x^2(3v + v^2)dx + x^2(1 + v)(vdx + xdv) = 0 \]
Dividing through by \(x^2\) (assuming \(x \neq 0\)):
\[ (3v + v^2)dx + (1 + v)(vdx + xdv) = 0 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ (3v + v^2)dx + (1 + v)vdx + (1 + v)xdv = 0 \]
Combining the \(dx\) terms:
\[ (3v + v^2 + v + v^2)dx + (1 + v)xdv = 0 \]
This simplifies to:
\[ (4v + 2v^2)dx + (1 + v)xdv = 0 \]

### Step 6: Separate variables
Now, separate the variables:
\[ \frac{(1 + v)}{(4v + 2v^2)} dv = -\frac{dx}{x} \]

### Step 7: Integrate both sides
Integrate both sides:
\[ \int \frac{(1 + v)}{(2v(2 + v))} dv = -\int \frac{dx}{x} \]
Using partial fractions, we can simplify the left side.

### Step 8: Solve the integrals
After integrating, we will have:
\[ \text{Left Side: } \text{(result of integration)} = -\ln|x| + C \]

### Step 9: Substitute back for \(v\)
Recall that \(v = \frac{y}{x}\). Substitute back to get the final equation in terms of \(x\) and \(y\).

### Step 10: Final form
The final form of the solution will be:
\[ x^2(y^2 + 2xy) = C \]

### Final Answer
Thus, the solution of the differential equation is:
\[ x^2(y^2 + 2xy) = C \]