Show that the equations y(x) are solution to the Ordinary Differential Equations.

2. ax + y=0
y(x) = Ce=*
3. y' = xy
y(x) =Cel

Let's analyze both cases and verify whether the given functions \( y(x) \) are indeed solutions to the corresponding ordinary differential equations (ODEs).

### Problem 2:
The given ODE is:
\[
a x + y = 0
\]
The proposed solution is:
\[
y(x) = C e^{x}
\]

#### Step-by-Step Verification:
1. First, substitute \( y(x) = C e^{x} \) into the ODE.

The ODE becomes:
\[
a x + C e^{x} = 0
\]

2. Since this equation needs to hold for all \(x\), we see that it cannot be true for arbitrary \(x\) and constant \(C\), because the terms \( a x \) and \( C e^x \) are not comparable or cancelable. Thus, the proposed solution \( y(x) = C e^x \) **is not a solution** to this ODE.

#### Conclusion:
The function \( y(x) = C e^x \) does **not** satisfy the equation \( ax + y = 0 \).

---

### Problem 3:
The given ODE is:
\[
y' = xy
\]
The proposed solution is:
\[
y(x) = C e^{\frac{x^2}{2}}
\]

#### Step-by-Step Verification:
1. Differentiate \( y(x) = C e^{\frac{x^2}{2}} \) with respect to \(x\).

\[
y'(x) = \frac{d}{dx} \left( C e^{\frac{x^2}{2}} \right) = C \cdot e^{\frac{x^2}{2}} \cdot \frac{d}{dx} \left( \frac{x^2}{2} \right) = C \cdot e^{\frac{x^2}{2}} \cdot x
\]

Thus, we have:
\[
y'(x) = x C e^{\frac{x^2}{2}}
\]

2. Now, substitute \( y(x) = C e^{\frac{x^2}{2}} \) and \( y'(x) = x C e^{\frac{x^2}{2}} \) into the original ODE \( y' = xy \).

\[
x C e^{\frac{x^2}{2}} = x \cdot C e^{\frac{x^2}{2}}
\]

Both sides of the equation are equal, which confirms that the proposed solution is indeed correct.

#### Conclusion:
The function \( y(x) = C e^{\frac{x^2}{2}} \) **is a solution** to the ODE \( y' = xy \).

---

### Summary:
1. For \( ax + y = 0 \), the function \( y(x) = C e^x \) is **not a solution**.
2. For \( y' = xy \), the function \( y(x) = C e^{\frac{x^2}{2}} \) is **a solution**.

---

Would you like further details on this, or do you have any questions?

Here are 5 related questions to explore:
1. How do you verify if a function is a solution to an ODE in general?
2. What are the general methods for solving first-order linear ODEs?
3. How does separation of variables work for solving ODEs?
4. Can we solve second-order differential equations with a similar method?
5. How does the constant \(C\) arise in the solutions of these ODEs?

**Tip:** When solving ODEs, always differentiate the proposed solution and substitute back into the equation to verify correctness.

Show that the equations y(x) are solutions to the Ordinary Differential Equations.
2. ax + y = 0, y(x) = Ce^x
3. y' = xy, y(x) = Ce^(x^2/2)

Explore how to verify solutions to Ordinary Differential Equations, focusing on examples like ax + y = 0 and y' = xy. This guide provides step-by-step verification using exponential functions and differentiation. Suitable for first-year calculus students or anyone studying differential equations.

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