Let y(x) be a curve given by differential equation `(dy)/(dx) - y = 1 + 4 sin x`. If `y(0)=1` then value of `y(pi/2)` is equal to

Options

A. `-3 + 4e^(pi/2)`
B. `3 - 4 e^(pi/2)`
C. `3 + 4 e^((-pi)/2)`
D. `3 + 4 e^(pi/2)`

Answer

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

### Step 1: Identify the integrating factor
The given differential equation is in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x) = -1\) and \(Q(x) = 1 + 4 \sin x\).

The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int P(x) \, dx} = e^{\int -1 \, dx} = e^{-x} \]

### Step 2: Multiply the entire equation by the integrating factor
Multiplying the differential equation by the integrating factor:
\[ e^{-x} \frac{dy}{dx} - e^{-x} y = e^{-x}(1 + 4 \sin x) \]

### Step 3: Rewrite the left-hand side
The left-hand side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(e^{-x} y) = e^{-x}(1 + 4 \sin x) \]

### Step 4: Integrate both sides
Now, we integrate both sides:
\[ \int \frac{d}{dx}(e^{-x} y) \, dx = \int e^{-x}(1 + 4 \sin x) \, dx \]
The left-hand side simplifies to:
\[ e^{-x} y \]
For the right-hand side, we can split the integral:
\[ \int e^{-x} \, dx + 4 \int e^{-x} \sin x \, dx \]

The first integral is:
\[ -e^{-x} \]

For the second integral, we can use integration by parts or the known result:
\[ \int e^{-x} \sin x \, dx = \frac{1}{2} e^{-x} (\sin x - \cos x) \]
Thus, we have:
\[ 4 \int e^{-x} \sin x \, dx = 2 e^{-x} (\sin x - \cos x) \]

Combining these results, we get:
\[ e^{-x} y = -e^{-x} + 2 e^{-x} (\sin x - \cos x) + C \]

### Step 5: Solve for \(y\)
Multiplying through by \(e^{x}\) to isolate \(y\):
\[ y = -1 + 2(\sin x - \cos x) + Ce^{x} \]

### Step 6: Apply the initial condition
Using the initial condition \(y(0) = 1\):
\[ 1 = -1 + 2(\sin 0 - \cos 0) + Ce^{0} \]
This simplifies to:
\[ 1 = -1 + 2(0 - 1) + C \]
\[ 1 = -1 - 2 + C \implies C = 4 \]

### Step 7: Write the particular solution
Substituting \(C\) back into the equation for \(y\):
\[ y = -1 + 2(\sin x - \cos x) + 4e^{x} \]

### Step 8: Find \(y\) at \(x = \frac{\pi}{2}\)
Now we calculate \(y\) at \(x = \frac{\pi}{2}\):
\[ y\left(\frac{\pi}{2}\right) = -1 + 2\left(\sin\frac{\pi}{2} - \cos\frac{\pi}{2}\right) + 4e^{\frac{\pi}{2}} \]
\[ = -1 + 2(1 - 0) + 4e^{\frac{\pi}{2}} = -1 + 2 + 4e^{\frac{\pi}{2}} = 1 + 4e^{\frac{\pi}{2}} \]

### Final Answer
Thus, the value of \(y\left(\frac{\pi}{2}\right)\) is:
\[ \boxed{1 + 4e^{\frac{\pi}{2}}} \]