Homogenous linear ODE with constant coëfficients

In the majority of cases solving a homogenous linear ODE with constant coëfficients

𝑎 0 𝑦 + 𝑎 1 𝑦 (1) + 𝑎 2 𝑦 (2) + \dots + 𝑦 (𝑛) = 0

equates to solving the characteristic equation

𝑎 0 + 𝑎 1 𝑟 + 𝑎 2 𝑟 2 + \dots + 𝑎 𝑛 𝑟 𝑛 = 0

where for any such $𝑟$ a solution is given by $𝑦 = 𝑒 𝑟 𝑥$ . In the cases where there are repeated roots, a method such as reduction of order must be used to find a complete basis of solutions.

Second order

In the second order case

𝑦 ″ + 𝑎 𝑦' + 𝑏 𝑦 = 0

the characteristic polynomial is

𝑟 2 + 𝑎 𝑟 + 𝑏 = 0

and a solution is given by

𝑦 = 𝐶 1 𝑒 𝑟 1 𝑥 + 𝐶 2 𝑒 𝑟 2 𝑥

If $𝑟 1 = 𝑟 2$ (repeated roots), then an additional linearly independent solution is

𝑦 = 𝐶 2 𝑥 𝑒 𝑟 𝑥

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Homogenous linear ODE with constant coëfficients

Second order

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