The Solution of the equation `(dy)/(dx)+2y=sin x` is

Options

A. `y=5(2sin x-cos x)+ce^(-2x)`
B. `y=(2 sin x -cos x)+ce^(-x)`
C. `y=(2cos x-sin x)+ce^(-2x)`
D. `y=(1)/(5) (2 sin x-cos x)+ ce^(-2x)`

Answer

To solve the differential equation \(\frac{dy}{dx} + 2y = \sin x\), we will follow these steps:

### Step 1: Identify the form of the differential equation
The given equation is in the form of a linear first-order differential equation:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
where \(p(x) = 2\) and \(q(x) = \sin x\).

### Step 2: Find the integrating factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int p(x) \, dx} \]
Substituting \(p(x) = 2\):
\[ I(x) = e^{\int 2 \, dx} = e^{2x} \]

### Step 3: Multiply the entire differential equation by the integrating factor
We multiply the original equation by the integrating factor:
\[ e^{2x} \frac{dy}{dx} + 2e^{2x}y = e^{2x} \sin x \]

### Step 4: Rewrite the left-hand side as a derivative
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}(e^{2x}y) = e^{2x} \sin x \]

### Step 5: Integrate both sides
Now, we integrate both sides with respect to \(x\):
\[ \int \frac{d}{dx}(e^{2x}y) \, dx = \int e^{2x} \sin x \, dx \]
The left side simplifies to:
\[ e^{2x}y = \int e^{2x} \sin x \, dx \]

### Step 6: Solve the integral on the right-hand side
To solve \(\int e^{2x} \sin x \, dx\), we will use integration by parts or a known formula. The formula for \(\int e^{ax} \sin(bx) \, dx\) is:
\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) \]
Here, \(a = 2\) and \(b = 1\):
\[ \int e^{2x} \sin x \, dx = \frac{e^{2x}}{2^2 + 1^2} (2 \sin x - 1 \cos x) = \frac{e^{2x}}{5} (2 \sin x - \cos x) \]

### Step 7: Substitute back into the equation
Substituting back, we have:
\[ e^{2x}y = \frac{e^{2x}}{5} (2 \sin x - \cos x) + C \]
where \(C\) is the constant of integration.

### Step 8: Solve for \(y\)
Now, we divide both sides by \(e^{2x}\):
\[ y = \frac{1}{5} (2 \sin x - \cos x) + Ce^{-2x} \]

### Final Solution
Thus, the solution to the differential equation \(\frac{dy}{dx} + 2y = \sin x\) is:
\[ y = \frac{1}{5} (2 \sin x - \cos x) + Ce^{-2x} \]

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