The exercise you uploaded contains several differential equations that need to be solved using the separation of variables method. Let's go through the solutions step-by-step for each equation.

Problem 1:

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

Solution:

1. Rearrange to separate variables: $\frac{dy}{1 + y} = (1 - x) dx$

2. Integrate both sides: $\int \frac{1}{1 + y} dy = \int (1 - x) dx$ This gives: $\ln|1 + y| = x - \frac{x^2}{2} + C$

3. Solve for $y$: $1 + y = e^{x - \frac{x^2}{2} + C} = e^C \cdot e^{x - \frac{x^2}{2}}$ Thus, the general solution is: $y = C e^{x - \frac{x^2}{2}} - 1$

Problem 2:

$\frac{dy}{dx} = \frac{x^2}{1 - y^2}$

Solution:

1. Separate variables: $(1 - y^2) dy = x^2 dx$

2. Integrate both sides: $\int (1 - y^2) dy = \int x^2 dx$ This gives: $y - \frac{y^3}{3} = \frac{x^3}{3} + C$

3. This is the implicit solution for $y$.

Problem 3:

$y' = 2(xy + y)$ This can be simplified to: $y' = 2y(x + 1)$

Solution:

1. Separate variables: $\frac{dy}{y} = 2(x + 1) dx$

2. Integrate both sides: $\int \frac{1}{y} dy = \int 2(x + 1) dx$ This gives: $\ln|y| = x^2 + 2x + C$

3. Solve for $y$: $y = C e^{x^2 + 2x}$

Problem 4:

$y' = \frac{y \cos y}{1 + 2y^2}$

Solution:

1. Separate variables: $\frac{1 + 2y^2}{y \cos y} dy = dx$ This is a more complicated integral, and it might not have a simple closed-form solution, but it could be approached using special functions or numeric integration.

Problem 5:

$y' = 2 e^{-y} \cos x$

Solution:

1. Separate variables: $\frac{e^y}{2} dy = \cos x dx$

2. Integrate both sides: $\int \frac{e^y}{2} dy = \int \cos x dx$ This gives: $\frac{e^y}{2} = \sin x + C$

3. Solve for $y$: $e^y = 2(\sin x + C)$ Thus, the solution is: $y = \ln[2(\sin x + C)]$

Do you want detailed steps for any specific equation?

Here are five related questions to deepen your understanding:

1. How does the separation of variables method work for non-linear differential equations?
2. Can all differential equations be solved using separation of variables?
3. What is the significance of the constant $C$ in differential equations?
4. How can implicit solutions be interpreted geometrically?
5. What is the role of integration constants in finding the general solution?

Tip: Always check the domain of your solutions to ensure they match any given initial conditions or restrictions in the problem.