Discussion
Match the nature of the auxiliary roots with the complementary function. 1. Real and distinct 2. Real and repeated 3. Complex conjugate P. \( (C_{1}+C_{2}x)e^{mx} \) Q. \( C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x} \) R. \( e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x) \)
Answer: C
Distinct real roots give two independent exponentials. A repeated root gives only one, so the second solution carries a factor of \( x \). A conjugate pair combines into a real exponential envelope times an oscillation, the real part fixing the envelope and the imaginary part the frequency
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