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The image contains the following mathematical problem:

### Given:
The function \( y(x) \) satisfies the second-order linear differential equation:
\[
y'' - 2y' + ky = 0, \quad \text{where } 0 < k < 1.
\]

### Tasks:
1. Find the limits:
\[
\lim_{x \to -\infty} y(x), \quad \lim_{x \to -\infty} y'(x).
\]
2. Given the conditions \( y(1) = 1 \) and \( y'(1) = 1 \), find the value of the following integral:
\[
\int_{-\infty}^{1} (kx^2 + 4x + 2) y(x) \, dx.
\]

### Steps for solving:

1. **Solve the Differential Equation**:
   The characteristic equation for the given differential equation \( y'' - 2y' + ky = 0 \) is:
   \[
   r^2 - 2r + k = 0.
   \]
   Solving for \( r \) gives the roots \( r_1 \) and \( r_2 \), which determine the general solution of \( y(x) \).

2. **Analyze Asymptotic Behavior**:
   Find the behavior of \( y(x) \) and \( y'(x) \) as \( x \to -\infty \) based on the general solution.

3. **Solve for Constants**:
   Use the boundary conditions \( y(1) = 1 \) and \( y'(1) = 1 \) to solve for any constants in the general solution.

4. **Evaluate the Integral**:
   Once \( y(x) \) is determined, the given integral \( \int_{-\infty}^{1} (kx^2 + 4x + 2) y(x) \, dx \) can be computed by substituting the expression for \( y(x) \).

---

Would you like a step-by-step breakdown starting with the solution of the differential equation? Feel free to ask if you have any questions!

Here are 5 related questions for further practice:
1. How do you solve second-order differential equations with constant coefficients?
2. What is the significance of the characteristic equation in solving differential equations?
3. How do boundary conditions affect the solution of differential equations?
4. How can the method of undetermined coefficients be used in non-homogeneous equations?
5. What are some methods for solving definite integrals involving exponential functions?

### Tip:
When solving second-order linear differential equations, always first check if the equation is homogeneous, as this helps in determining the general form of the solution.

设函数 y(x) 满足方程 y'' - 2y' + ky = 0, 其中 0 < k < 1，求解相关问题。

Explore how to solve the second-order differential equation y'' - 2y' + ky = 0, where 0 < k < 1. This problem includes finding limits at infinity and evaluating a definite integral. Suitable for undergraduate students studying differential equations and calculus.

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