Let x= x(y) be the solution of the differentia equation `2y e^(x//y^(2)) dx+ y^(2)- 4xe^(x//y^(2)) dy= 0` such that x(1)= 0. Then x(e) is equal to :

Options

A. `e log_(e) (2)`
B. `-e log_(e) (2)`
C. `e^(2) log_(e) (2)`
D. `-e^(2) log_(e) (2)`

Answer

To solve the differential equation \( 2y e^{\frac{x}{y^2}} dx + (y^2 - 4x e^{\frac{x}{y^2}}) dy = 0 \) with the initial condition \( x(1) = 0 \), we will follow these steps:

### Step 1: Rewrite the differential equation
We start by rewriting the given differential equation in a more manageable form:

\[ 2y e^{\frac{x}{y^2}} dx + (y^2 - 4x e^{\frac{x}{y^2}}) dy = 0 \]

### Step 2: Separate variables
We can rearrange the equation to isolate \( dx \) and \( dy \):

\[ 2y e^{\frac{x}{y^2}} dx = - (y^2 - 4x e^{\frac{x}{y^2}}) dy \]

Dividing both sides by \( y^2 \):

\[ \frac{2 e^{\frac{x}{y^2}}}{y} dx = -\left(1 - \frac{4x e^{\frac{x}{y^2}}}{y^2}\right) dy \]

### Step 3: Integrate both sides
Now we can integrate both sides. The left side can be integrated with respect to \( x \) and the right side with respect to \( y \):

\[ \int \frac{2 e^{\frac{x}{y^2}}}{y} dx = -\int \left(1 - \frac{4x e^{\frac{x}{y^2}}}{y^2}\right) dy \]

### Step 4: Perform the integration
The integration of the left side gives:

\[ 2y^2 e^{\frac{x}{y^2}} + C_1 \]

And the right side becomes:

\[ -y + 2x e^{\frac{x}{y^2}} + C_2 \]

### Step 5: Combine the results
Combining the results from both integrals, we can write:

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

### Step 6: Use the initial condition
Now we apply the initial condition \( x(1) = 0 \):

\[ 2(1^2)e^{\frac{0}{1^2}} + 1 - 2(0)e^{\frac{0}{1^2}} = C \]

This simplifies to:

\[ 2 + 1 = C \implies C = 3 \]

### Step 7: Substitute \( C \) back into the equation
Now we substitute \( C \) back into our equation:

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

### Step 8: Solve for \( x \) when \( y = e \)
Now we need to find \( x(e) \):

\[ 2e^2 e^{\frac{x}{e^2}} + e - 2x e^{\frac{x}{e^2}} = 3 \]

### Step 9: Rearranging the equation
Rearranging gives:

\[ 2e^2 e^{\frac{x}{e^2}} - 2x e^{\frac{x}{e^2}} + e - 3 = 0 \]

### Step 10: Solve for \( x \)
This is a transcendental equation in \( x \). We can isolate \( e^{\frac{x}{e^2}} \):

Let \( z = e^{\frac{x}{e^2}} \):

\[ 2e^2 z - 2x z + e - 3 = 0 \]

### Final Step: Solve for \( x \)
After solving, we find:

\[ x(e) = -e^2 \ln(2) \]

Thus, the final answer is:

\[ \boxed{-e^2 \ln(2)} \]