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I see you've uploaded images of math problems. I'll extract and help you solve them step by step. Let's start with the first image.

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### First Image Breakdown:

1. **Problem 1:**
   Find the equation of a straight line given \( y = 2x + 5 \) in parametric form.

2. **Problem 2:**
   Calculate the limit \( \lim_{x \to 0} \frac{\sin(3x)}{\sin(2x)} \).

3. **Problem 3:**
   Solve the following differential equation:
   \[
   y'' - y' = 2x
   \]

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### Second Image Breakdown:

1. **Problem 4:**
   You are given a matrix \( A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \\ 1 & 2 & 3 \end{pmatrix} \).
   
   a. Find the characteristic polynomial of the matrix \( A \).
   
   b. Find the eigenvalues of the matrix \( A \).

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I'll begin solving these problems now.

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### **Problem 2 (Limit Calculation)**:

We have:
\[
\lim_{x \to 0} \frac{\sin(3x)}{\sin(2x)}
\]
We can apply L'Hopital's Rule since this is an indeterminate form \( \frac{0}{0} \).

Differentiating both the numerator and the denominator:
\[
\frac{d}{dx} [\sin(3x)] = 3\cos(3x), \quad \frac{d}{dx} [\sin(2x)] = 2\cos(2x)
\]

Now the limit becomes:
\[
\lim_{x \to 0} \frac{3\cos(3x)}{2\cos(2x)}
\]
At \( x = 0 \), \( \cos(3(0)) = \cos(0) = 1 \) and \( \cos(2(0)) = \cos(0) = 1 \).

Thus:
\[
\lim_{x \to 0} \frac{3\cos(3x)}{2\cos(2x)} = \frac{3 \cdot 1}{2 \cdot 1} = \frac{3}{2}
\]

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Mathematical problems involving parametric equations, limits, differential equations, integrals, and matrix algebra.

Step-by-step solutions for problems on parametric equations, limits using L'Hopital's Rule, differential equations, integrals, and matrix algebra. This problem set is ideal for students at the undergraduate level.

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