If the solution of
\[
\left( 1 + 2e^\frac{x}{y} \right) dx + 2e^\frac{x}{y} \left( 1 - \frac{x}{y} \right) dy = 0
\]
is
\[
x + \lambda y e^\frac{x}{y} = c \quad \text{(where \(c\) is an arbitrary constant), then \( \lambda \) is:}
\]
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When solving differential equations, identifying patterns in the given equation can help choose a useful substitution that simplifies the problem.