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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.

Tentukan penyelesaian dari persamaan diferensial berikut dengan metode variabel terpisah

Learn how to solve differential equations using the separation of variables method. This tutorial covers five equations, each with detailed step-by-step solutions, suitable for undergraduate or advanced high school students.

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