Solve the following differential equations
`(dy)/(dx)+(1)/(x)y = cos x+(sin x)/(x), " " x gt 0`.

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{1}{x}y = \cos x + \frac{\sin x}{x}, \quad x > 0, \]

we will follow the steps for solving a linear first-order differential equation.

### Step 1: Identify p and q

The given equation is in the standard form

\[ \frac{dy}{dx} + p(x)y = q(x), \]

where

\[ p(x) = \frac{1}{x} \quad \text{and} \quad q(x) = \cos x + \frac{\sin x}{x}. \]

### Step 2: Find the Integrating Factor

The integrating factor \( \mu(x) \) is given by

\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx}. \]

Calculating the integral:

\[ \int \frac{1}{x} \, dx = \ln |x| \implies \mu(x) = e^{\ln |x|} = |x| = x \quad \text{(since \( x > 0 \))}. \]

### Step 3: Multiply the Differential Equation by the Integrating Factor

Now, we multiply the entire differential equation by the integrating factor \( x \):

\[ x \frac{dy}{dx} + y = x \cos x + \sin x. \]

### Step 4: Rewrite the Left Side

The left side can be rewritten as the derivative of a product:

\[ \frac{d}{dx}(xy) = x \cos x + \sin x. \]

### Step 5: Integrate Both Sides

Now, we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx}(xy) \, dx = \int (x \cos x + \sin x) \, dx. \]

The left side simplifies to:

\[ xy = \int (x \cos x) \, dx + \int \sin x \, dx. \]

### Step 6: Solve the Integrals

1. **Integrate \( x \cos x \)** using integration by parts:
- Let \( u = x \) and \( dv = \cos x \, dx \).
- Then, \( du = dx \) and \( v = \sin x \).
- Applying integration by parts:

\[ \int x \cos x \, dx = x \sin x - \int \sin x \, dx = x \sin x + \cos x. \]

2. **Integrate \( \sin x \)**:

\[ \int \sin x \, dx = -\cos x. \]

### Step 7: Combine the Results

Now substituting back into the equation:

\[ xy = (x \sin x + \cos x) - \cos x + C, \]

where \( C \) is the constant of integration. Thus,

\[ xy = x \sin x + C. \]

### Step 8: Solve for y

Finally, we can solve for \( y \):

\[ y = \sin x + \frac{C}{x}. \]

### Final Solution

The solution to the differential equation is:

\[ y = \sin x + \frac{C}{x}. \]

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