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Let's begin by breaking down the problems shown in the image. There are two main tasks outlined:

### 1. Solve the Differential Equations with Initial Conditions
These are first-order and second-order differential equations where particular solutions are required using initial values.

- **(a)** \( \frac{dy}{dx} = \frac{x}{y+1} \), \( y(0) = 1 \)
- **(b)** \( \frac{dy}{dx} = x(2y^2 - xdy) \), \( y(1) = 4 \)
- **(c)** \( \frac{dy}{dx} = 2y \cdot \cot(x) \), \( y\left(\frac{\pi}{2}\right) = 2 \)
- **(d)** \( (xy^2 + x)dx + (x^2y - y)dy = 0 \), \( y(0) = 1 \)

### 2. Find the General Solutions of the Homogeneous Linear Differential Equations with Constant Coefficients
These equations involve higher-order differential operators and require solving characteristic equations.

- **(a)** \( \frac{d^2y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0 \)
- **(b)** \( \frac{d^2y}{dx^2} + 4y = 0 \)
- **(c)** \( \frac{d^2y}{dx^2} + y = 0 \)
- **(d)** \( \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + y = 0 \)
- **(e)** \( \frac{d^3y}{dx^3} - 2 \frac{d^2y}{dx^2} + 2y = 0 \)
- **(f)** \( \frac{d^4y}{dx^4} - y = 0 \)
- **(g)** \( \frac{d^3y}{dx^3} - y = 0 \)
- **(h)** \( \frac{d^3y}{dx^3} - 3 \frac{d^2y}{dx^2} - y = 0 \)
- **(i)** \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0 \)
- **(j)** \( \frac{d^4y}{dx^4} - 4 \frac{d^3y}{dx^3} + 6 \frac{d^2y}{dx^2} - 4 \frac{dy}{dx} + y = 0 \)

---

I will begin solving the first problem in part 1(a) and part 2(a).

---

### 1. (a) Solving \( \frac{dy}{dx} = \frac{x}{y+1} \), \( y(0) = 1 \)

This is a separable differential equation:

\[
\frac{dy}{dx} = \frac{x}{y + 1}
\]

To solve, rearrange to separate variables:

\[
(y + 1) \, dy = x \, dx
\]

Integrating both sides:

\[
\int (y + 1) \, dy = \int x \, dx
\]

\[
\frac{y^2}{2} + y = \frac{x^2}{2} + C
\]

Apply the initial condition \( y(0) = 1 \):

\[
\frac{(1)^2}{2} + 1 = \frac{(0)^2}{2} + C
\]

\[
\frac{1}{2} + 1 = C \quad \Rightarrow \quad C = \frac{3}{2}
\]

Thus, the solution is:

\[
\frac{y^2}{2} + y = \frac{x^2}{2} + \frac{3}{2}
\]

Now, I will move on to solving part 2(a).

---

### 2. (a) Solving \( \frac{d^2y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0 \)

This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is:

\[
r^2 - 3r + 2 = 0
\]

Factoring:

\[
(r - 1)(r - 2) = 0
\]

Thus, the solutions to the characteristic equation are \( r = 1 \) and \( r = 2 \). Therefore, the general solution to the differential equation is:

\[
y(x) = C_1 e^x + C_2 e^{2x}
\]

---

Would you like a detailed explanation of any steps, or should I continue solving the remaining problems?

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### Related Questions:

1. How do you approach solving non-homogeneous differential equations?
2. What are characteristic equations in solving higher-order differential equations?
3. How do initial conditions influence the general solution of a differential equation?
4. What is the method of undetermined coefficients used for solving particular solutions?
5. How do you solve systems of differential equations with multiple unknowns?

### Tip:
When solving separable differential equations, always ensure to properly integrate both sides and don’t forget to apply initial conditions after integration to find the constant of integration.

Solve the differential equations and find the general solutions for linear homogeneous differential equations with constant coefficients.

Learn how to solve differential equations with initial conditions and find general solutions for homogeneous linear differential equations. Includes step-by-step solutions and explanations suitable for undergraduate economics and mathematics students.

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