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Interview prep / Pure Math

General solution of y'' + y' + y = 0

Question

Find all solutions of y′′+y′+y=0y''+y'+y=0.

Short answer

y=e−x/2(c1cos⁡32x+c2sin⁡32x)y=e^{-x/2}\left(c_1\cos\tfrac{\sqrt3}{2}x+c_2\sin\tfrac{\sqrt3}{2}x\right). The characteristic equation r2+r+1=0r^2+r+1=0 has complex roots −12±32i-\tfrac12\pm\tfrac{\sqrt3}{2}i; real part sets the decay, imaginary part the oscillation.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 3.5 Ordinary Differential Equations, 'Homogeneous linear ODE', p.48; question and answer rewritten by SelfMadeQuant.com.

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