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 $ax + y = 0$.

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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 $y' = xy$.

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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.

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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.