We are given the differential equation:

$y' = (1 - y) \cos x$

This is a first-order non-linear differential equation. To solve it, we can use a method called separation of variables, where we try to separate the variables $y$ and $x$ on opposite sides of the equation. Let's proceed with the steps.

Step 1: Separate the variables

We start by rewriting the equation as:

$\frac{dy}{dx} = (1 - y) \cos x$

Now, divide both sides by $1 - y$ and multiply both sides by $dx$ to get:

$\frac{1}{1 - y} \, dy = \cos x \, dx$

Step 2: Integrate both sides

Next, integrate both sides:

$\int \frac{1}{1 - y} \, dy = \int \cos x \, dx$

• The integral on the left-hand side is:

$\int \frac{1}{1 - y} \, dy = -\ln|1 - y|$

• The integral on the right-hand side is:

$\int \cos x \, dx = \sin x$

Thus, we get:

$-\ln|1 - y| = \sin x + C$

where $C$ is the constant of integration.

Step 3: Solve for $y$

Now, solve for $y$. First, multiply both sides by $-1$:

$\ln|1 - y| = -\sin x - C$

Take the exponential of both sides:

$|1 - y| = e^{-\sin x - C} = e^{-C} e^{-\sin x}$

Let $A = e^{-C}$, which is a constant. We now have:

$|1 - y| = A e^{-\sin x}$

Finally, remove the absolute value and solve for $y$:

$1 - y = A e^{-\sin x}$ $y = 1 - A e^{-\sin x}$

Final Solution:

The general solution to the differential equation is:

$y(x) = 1 - A e^{-\sin x}$

where $A$ is an arbitrary constant determined by initial conditions.

Would you like more details on any step or a clarification?

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Here are 5 related questions for further exploration:

1. What is the method of separation of variables, and when is it applicable?
2. How does the integration constant affect the solution of a differential equation?
3. Can this equation be solved using an alternate method like an integrating factor?
4. What are the properties of exponential and logarithmic functions in solving differential equations?
5. How can initial conditions be applied to determine the constant $A$?

Tip: When solving first-order differential equations, always check if the equation is separable; this method is one of the simplest and most effective when applicable.