`y=y(x)` is the solution of the differential equation `2x^2dy/dx-2xy+8y^2=0 , y(e)=e/3` , then `y(1)` is equal to

Options

A. `2/3`
B. 3
C. `3/2`
D. -1

Answer

To solve the differential equation \( 2x^2 \frac{dy}{dx} - 2xy + 8y^2 = 0 \) with the initial condition \( y(e) = \frac{e}{3} \) and find \( y(1) \), we can follow these steps:

### Step 1: Rearranging the Differential Equation
We start with the given equation:
\[ 2x^2 \frac{dy}{dx} - 2xy + 8y^2 = 0 \]
We can rearrange this to isolate \( \frac{dy}{dx} \):
\[ 2x^2 \frac{dy}{dx} = 2xy - 8y^2 \]
Dividing through by \( 2x^2 \):
\[ \frac{dy}{dx} = \frac{2xy - 8y^2}{2x^2} = \frac{xy - 4y^2}{x^2} \]

### Step 2: Separating Variables
We can separate the variables by rewriting the equation:
\[ \frac{dy}{y^2} = \frac{(x - 4y)}{x^2} dx \]
This can be rearranged to:
\[ \frac{1}{y^2} dy = \left( \frac{1}{x} - \frac{4}{x^2} \right) dx \]

### Step 3: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{1}{y^2} dy = \int \left( \frac{1}{x} - \frac{4}{x^2} \right) dx \]
The left side integrates to:
\[ -\frac{1}{y} = \ln |x| + \frac{4}{x} + C \]

### Step 4: Solving for \( y \)
Rearranging gives us:
\[ \frac{1}{y} = -\ln |x| - \frac{4}{x} - C \]
Thus,
\[ y = \frac{1}{- \ln |x| - \frac{4}{x} - C} \]

### Step 5: Applying the Initial Condition
We use the initial condition \( y(e) = \frac{e}{3} \):
\[ \frac{1}{\frac{e}{3}} = -\ln |e| - \frac{4}{e} - C \]
This simplifies to:
\[ \frac{3}{e} = -1 - \frac{4}{e} - C \]
Rearranging gives:
\[ C = -1 - \frac{4}{e} - \frac{3}{e} = -1 - \frac{7}{e} \]

### Step 6: Finding \( y(1) \)
Now we substitute \( C \) back into the equation for \( y \):
\[ y = \frac{1}{- \ln |x| - \frac{4}{x} + 1 + \frac{7}{e}} \]
To find \( y(1) \):
\[ y(1) = \frac{1}{- \ln(1) - 4 + 1 + \frac{7}{e}} = \frac{1}{0 - 4 + 1 + \frac{7}{e}} = \frac{1}{-3 + \frac{7}{e}} \]

### Step 7: Simplifying \( y(1) \)
Calculating \( y(1) \):
\[ y(1) = \frac{1}{\frac{7}{e} - 3} \]
This gives us the final answer for \( y(1) \).

### Final Answer
Thus, the value of \( y(1) \) is:
\[ y(1) = \frac{e}{7 - 3e} \]