To verify whether the given differential equation is exact and then find its general solution, let's first analyze the given equation step by step.

Given Differential Equation:

$y'' + x y' + y = 0$

This is a second-order linear differential equation. To determine whether it's exact, we first need to express it in a form that allows us to apply the test for exactness. Typically, we check exactness for first-order differential equations, but this equation can be solved by finding an appropriate method (such as reduction of order or using an auxiliary equation), rather than testing exactness directly.

Let's approach solving it instead, as testing exactness is not typically done for second-order linear DEs.

Step 1: Assume a solution of the form $y = e^{rx}$

This is a standard approach for solving second-order linear homogeneous differential equations with constant coefficients.

If $y = e^{rx}$, then: $y' = r e^{rx}, \quad y'' = r^2 e^{rx}$

Substitute these into the original equation: $r^2 e^{rx} + x (r e^{rx}) + e^{rx} = 0$

Factor out $e^{rx}$ (which is non-zero): $e^{rx} \left( r^2 + xr + 1 \right) = 0$

For non-trivial solutions, the expression inside the parentheses must be zero: $r^2 + xr + 1 = 0$

Step 2: Solve for $r$

The term $r^2 + xr + 1 = 0$ is a non-trivial quadratic-like expression that doesn't fit into the typical form for solving directly with constant coefficients. Since we see a linear term in $x$, we conclude that standard methods like constant coefficients don't apply directly. Thus, this suggests the equation may be better approached by series solutions or other methods (as it's nonlinear in this case).

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Would you like me to attempt an alternative method, such as solving via power series or using a known transformation? Also, feel free to ask for clarification on any steps.

Here are 5 related questions for deeper understanding:

1. What are the different methods to solve second-order linear DEs?
2. How do we apply the power series method to solve differential equations?
3. What are exact equations, and how do they apply to first-order DEs?
4. When can we use the method of undetermined coefficients in solving DEs?
5. How do auxiliary equations help solve second-order homogeneous DEs?

Tip: When solving second-order linear DEs, check if the coefficients are constant; if not, consider transformations or series solutions.