To solve the differential equation \(\frac{dy}{dx} = (1-y)\cos x\) with the initial condition \(y(\pi) = 2\), we can follow these steps:

### Step 1: Separate the Variables
Separate the variables to express the equation in the form where all terms involving \(y\) are on one side and all terms involving \(x\) are on the other:

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

### Step 2: Integrate Both Sides
Integrate both sides of the equation:

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

The left side integrates to:

\[
-\ln |1-y| = \sin x + C
\]

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

### Step 3: Solve for \(y\)
To solve for \(y\), first multiply by \(-1\):

\[
\ln |1-y| = -\sin x - C
\]

Exponentiate both sides to remove the logarithm:

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

Let \(e^{-C} = K\), where \(K\) is a new constant:

\[
1-y = \pm K e^{-\sin x}
\]

Solve for \(y\):

\[
y = 1 \mp K e^{-\sin x}
\]

### Step 4: Apply the Initial Condition
Use the initial condition \(y(\pi) = 2\) to find the constant \(K\):

\[
2 = 1 \mp K e^{-\sin \pi}
\]

Since \(\sin \pi = 0\):

\[
2 = 1 \mp K
\]

Thus:

\[
K = 1
\]

### Step 5: Write the Final Solution
Substituting \(K = 1\) into the solution:

\[
y = 1 - e^{-\sin x}
\]

This is the final solution to the differential equation.

Would you like more details or have any questions?

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Here are some related questions to consider:

1. How would the solution change if the initial condition was \(y(0) = 2\)?
2. What happens to the solution as \(x\) approaches infinity?
3. Can this differential equation be solved using an alternative method, such as the integrating factor method?
4. How would you interpret the physical meaning of this solution if it represented a population model?
5. What if the differential equation was \(\frac{dy}{dx} = (1-y)\sin x\) instead?

**Tip:** Always check the initial condition after finding the general solution to ensure that the solution is correct.

Learn how to solve the differential equation dy/dx = (1-y)cosx with initial condition y(π) = 2. This problem involves separating variables, integrating both sides, and applying an initial condition. Suitable for advanced undergraduate students and those interested in differential equations.

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