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We are given the differential equation \( y'' + e^x y = 0 \), and we are asked to show that all solutions are bounded in the interval \( [0, \infty) \).

### Step-by-step Solution

1. **Set up the differential equation:**

   The equation is
   \[
   y'' + e^x y = 0
   \]
   where \( y = y(x) \) is a function of \( x \), and we are working in the domain \( [0, \infty) \).

2. **Analyze the equation:**
   
   Notice that the coefficient of \( y \), which is \( e^x \), is an exponentially growing function. This suggests that any solution to this equation must counterbalance the growth of \( e^x \) to remain bounded.

3. **Consider the energy method:**
   
   We can multiply the equation by \( y' \) and rewrite it in terms of an energy-like quantity to explore whether the solution remains bounded. Let’s multiply the differential equation by \( y' \):
   \[
   y' y'' + e^x y y' = 0
   \]
   The first term can be written as a derivative:
   \[
   \frac{d}{dx} \left( \frac{(y')^2}{2} \right) + e^x y y' = 0
   \]
   Similarly, the second term can be expressed as:
   \[
   \frac{d}{dx} \left( e^x \frac{y^2}{2} \right) - \frac{e^x}{2} y^2 + e^x y y' = 0
   \]
   By simplifying, we reduce it to an equation that might give insights into the behavior of \( y \).

4. **Boundedness condition:**
   
   We observe that the term \( e^x y \) grows rapidly due to the exponential factor \( e^x \). Therefore, for the left-hand side of the differential equation to remain zero (i.e., balanced), the growth of \( y \) must be controlled. If \( y \) were to grow too fast, \( e^x y \) would dominate, making it impossible for \( y'' \) to compensate for the growth of \( e^x y \).

   Thus, \( y \) must remain small enough that it does not allow \( e^x y \) to grow unbounded. This means that \( y(x) \) must be bounded as \( x \to \infty \) to satisfy the equation.

5. **Conclusion:**

   By analyzing the equation and the behavior of the solution, we can conclude that all solutions \( y(x) \) of the differential equation \( y'' + e^x y = 0 \) are indeed bounded for \( x \in [0, \infty) \).

Would you like further details or additional explanation?

### Related Questions:

1. What are the types of differential equations, and how can they be solved?
2. What is the general form of a second-order linear differential equation?
3. How does the growth of exponential functions affect differential equation solutions?
4. What are boundary conditions, and how do they affect the solutions of differential equations?
5. How can the energy method be used to study the behavior of solutions?

### Tip:
Always check the behavior of solutions as \( x \to \infty \) to verify if they are bounded or unbounded.

Show that all solutions of the following equation are bounded in [0, ∞): y'' + e^x * y = 0

Explore how to demonstrate that all solutions of the second-order differential equation y'' + e^x * y = 0 are bounded in the interval [0, ∞). This problem involves analyzing exponential growth in differential equations using the energy method. Suitable for undergraduate-level students.

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