Solve the following differential equations
`(dy)/(dx)-2y= cos 3x`.

Answer

To solve the differential equation

\[ \frac{dy}{dx} - 2y = \cos(3x), \]

we will follow these steps:

### Step 1: Identify the type of differential equation
This is a first-order linear differential equation of the form

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

where \( p(x) = -2 \) and \( q(x) = \cos(3x) \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by

\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int -2 \, dx} = e^{-2x}. \]

### Step 3: Multiply the entire equation by the integrating factor
Multiply both sides of the original equation by \( e^{-2x} \):

\[ e^{-2x} \frac{dy}{dx} - 2e^{-2x}y = e^{-2x} \cos(3x). \]

This simplifies to:

\[ \frac{d}{dx}(e^{-2x}y) = e^{-2x} \cos(3x). \]

### Step 4: Integrate both sides
Now we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx}(e^{-2x}y) \, dx = \int e^{-2x} \cos(3x) \, dx. \]

The left side simplifies to:

\[ e^{-2x}y. \]

For the right side, we need to solve the integral \( \int e^{-2x} \cos(3x) \, dx \). We will use integration by parts or the method of undetermined coefficients.

Let \( I = \int e^{-2x} \cos(3x) \, dx \).

Using integration by parts, we can let:

- \( u = \cos(3x) \) and \( dv = e^{-2x}dx \)
- Then, \( du = -3\sin(3x)dx \) and \( v = -\frac{1}{2}e^{-2x} \)

Applying integration by parts:

\[ I = -\frac{1}{2} \cos(3x)e^{-2x} - \int -\frac{1}{2}(-3\sin(3x)e^{-2x})dx. \]

This gives:

\[ I = -\frac{1}{2} \cos(3x)e^{-2x} + \frac{3}{2} \int e^{-2x} \sin(3x)dx. \]

Let \( J = \int e^{-2x} \sin(3x)dx \). We can apply integration by parts again on \( J \):

Using similar steps, we can find \( J \) and substitute it back into the equation for \( I \).

### Step 5: Solve for \( y \)
After finding \( I \), we can express:

\[ e^{-2x}y = I + C, \]

where \( C \) is the constant of integration. Finally, we multiply both sides by \( e^{2x} \):

\[ y = e^{2x}(I + C). \]

### Final Solution
After performing the integrations and substituting back, we will arrive at the final solution for \( y \).