solve the following de cos(x)y'+sin(x)y=1

We are given the differential equation:

\[
\cos(x) \frac{dy}{dx} + \sin(x) y = 1
\]

To solve this first-order linear differential equation, we'll use an integrating factor approach.

### Step 1: Rewrite the equation in standard form

We start by getting the equation into the form:

\[
\frac{dy}{dx} + P(x) y = Q(x)
\]

Dividing through by \(\cos(x)\), we get:

\[
\frac{dy}{dx} + \frac{\sin(x)}{\cos(x)} y = \frac{1}{\cos(x)}
\]

This simplifies to:

\[
\frac{dy}{dx} + \tan(x) y = \sec(x)
\]

Now, the equation is in the standard form with:
- \( P(x) = \tan(x) \)
- \( Q(x) = \sec(x) \)

### Step 2: Find the integrating factor

The integrating factor is given by:

\[
\mu(x) = e^{\int P(x) dx} = e^{\int \tan(x) dx}
\]

The integral of \(\tan(x)\) is \(\ln|\sec(x)|\), so the integrating factor becomes:

\[
\mu(x) = e^{\ln|\sec(x)|} = \sec(x)
\]

### Step 3: Multiply through by the integrating factor

Now multiply the entire differential equation by \(\sec(x)\):

\[
\sec(x) \frac{dy}{dx} + \sec(x) \tan(x) y = \sec^2(x)
\]

This simplifies to:

\[
\frac{d}{dx} [y \sec(x)] = \sec^2(x)
\]

### Step 4: Integrate both sides

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

\[
y \sec(x) = \int \sec^2(x) dx
\]

The integral of \(\sec^2(x)\) is \(\tan(x)\), so:

\[
y \sec(x) = \tan(x) + C
\]

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

### Step 5: Solve for \(y\)

Finally, solve for \(y\):

\[
y = \cos(x) (\tan(x) + C)
\]

Simplifying:

\[
y = \sin(x) + C \cos(x)
\]

### Final Solution:

The solution to the differential equation is:

\[
y(x) = \sin(x) + C \cos(x)
\]

### Would you like further details or have any questions?

Here are five related questions that expand upon this concept:
1. How do you solve a second-order linear differential equation?
2. What are some real-life applications of differential equations?
3. Can we use a different method to solve this equation, such as variation of parameters?
4. How does the integrating factor method differ from separation of variables?
5. Can you solve a non-homogeneous differential equation using the same approach?

**Tip:** Always verify that the solution satisfies the original differential equation by substituting it back in!

Discover how to solve the first-order linear differential equation cos(x)y' + sin(x)y = 1 using the integrating factor method. This step-by-step solution breaks down key concepts such as trigonometric functions and integrating factors, suitable for undergraduate students.

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