If a curve follows the diffferential equation `dy/dx=(2^(x+y)-2^x)/2^y` and curve passes through the point (0,1) then the value of `y(2)` is

Options

A. `y=log_2(1-e^3)`
B. `y=log_2(1+e^3)`
C. `y=log_2(e^3-1)`
D. `y=log_2(e^-3+1)`

Answer

To solve the given differential equation \( \frac{dy}{dx} = \frac{2^{x+y} - 2^x}{2^y} \) and find the value of \( y(2) \) given that the curve passes through the point \( (0, 1) \), we can follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the differential equation:
\[ \frac{dy}{dx} = \frac{2^{x+y} - 2^x}{2^y} \]
This can be rewritten as:
\[ \frac{dy}{dx} = \frac{2^x \cdot 2^y - 2^x}{2^y} = 2^x \left(1 - \frac{1}{2^y}\right) \]

### Step 2: Separate Variables
We can separate the variables by multiplying both sides by \( 2^y \) and rearranging:
\[ 2^y dy = 2^x (1 - \frac{1}{2^y}) dx \]
This simplifies to:
\[ 2^y dy = 2^x dx - 2^{x-y} dx \]

### Step 3: Integrate Both Sides
Now we need to integrate both sides. We can rewrite the left-hand side as:
\[ \int 2^y dy = \int 2^x dx - \int 2^{x-y} dx \]
The left side integrates to:
\[ \frac{2^y}{\ln 2} \]
For the right side, we have:
\[ \int 2^x dx = \frac{2^x}{\ln 2} + C \]
And for \( \int 2^{x-y} dx \), we treat \( y \) as a constant:
\[ \int 2^{x-y} dx = \frac{2^{x-y}}{\ln 2} \]

### Step 4: Combine Integrals
Putting it all together, we have:
\[ \frac{2^y}{\ln 2} = \frac{2^x}{\ln 2} - \frac{2^{x-y}}{\ln 2} + C \]
Multiplying through by \( \ln 2 \) gives:
\[ 2^y = 2^x - 2^{x-y} + C \ln 2 \]

### Step 5: Use Initial Condition
Now we use the initial condition \( (0, 1) \):
\[ 2^1 = 2^0 - 2^{0-1} + C \ln 2 \]
This simplifies to:
\[ 2 = 1 - \frac{1}{2} + C \ln 2 \]
Thus:
\[ 2 = \frac{1}{2} + C \ln 2 \implies C \ln 2 = 2 - \frac{1}{2} = \frac{3}{2} \implies C = \frac{3}{2 \ln 2} \]

### Step 6: Substitute Back to Find \( y \)
Now substituting \( C \) back into our equation:
\[ 2^y = 2^x - 2^{x-y} + \frac{3}{2} \]

### Step 7: Find \( y(2) \)
Now we need to find \( y(2) \):
\[ 2^y = 2^2 - 2^{2-y} + \frac{3}{2} \]
This simplifies to:
\[ 2^y = 4 - 2^{2-y} + \frac{3}{2} \]
Combining terms gives:
\[ 2^y = \frac{8}{2} - \frac{3}{2} + \frac{3}{2} = 4 \]
Thus:
\[ 2^y = 4 \implies y = 2 \]

### Final Answer
The value of \( y(2) \) is:
\[ \boxed{2} \]