Characteristic Equation Method
Substituting y = e^(rt) into a constant-coefficient linear homogeneous ODE ay'' + by' + cy = 0 reduces it to the polynomial ar^2 + br + c = 0; distinct real roots give exponentials, a repeated root contributes a factor of t, and complex conjugate roots give exponentially weighted sines and cosines via Euler's formula. The general solution is a linear combination of these fundamental solutions.
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Substituting y = e^(rt) into a constant-coefficient linear homogeneous ODE ay'' + by' + cy = 0 reduces it to the polynomial ar^2 + br + c = 0; distinct real roots give exponentials, a repeated root contributes a factor of t, and complex conjugate roots give exponentially weighted sines and cosines via Euler's formula. The general solution is a linear combination of these fundamental solutions.
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