ODE2.2

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2y''+y'+4y=0\,

Set up and solve the characteristic equation.

2m^{2}+m+4=0\,

m={\frac  {-1\pm {\sqrt  {1-4\cdot 8}}}{4}}={\frac  {-1}{4}}\pm {\frac  {i{\sqrt  {31}}}{4}}\,

So the general solution is

y(x)=c_{1}e^{{-1/4x}}\cos \left({\frac  {{\sqrt  {31}}}{4}}x\right)+c_{2}e^{{-1/4x}}\sin \left({\frac  {{\sqrt  {31}}}{4}}x\right)\,

Ordinary Differential Equations

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