Ordinary Differential Equations

# Ordinary Differential Equations - Tuesday Ordinary...

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Tuesday, October 27, 2009 1 Ordinary Differential Equations (ODEs) Linear 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑑𝑑 ̇ = 𝐴𝐴 ( 𝑑𝑑 ) 𝑑𝑑 + 𝑏𝑏 ( 𝑑𝑑 ) 𝑑𝑑̇ = 𝑓𝑓 ( 𝑑𝑑 , 𝑑𝑑 ) Non Linear 𝑑𝑑 (0) = 𝑑𝑑 ° 𝑑𝑑 (0) = 𝑑𝑑 ° Usual substitution 𝑑𝑑 ̇ = 𝐴𝐴𝑑𝑑 + 𝑏𝑏 ( 𝑑𝑑 ) This can be solved analytically 𝑑𝑑 ̇ = 𝐴𝐴𝑑𝑑 + 𝑏𝑏 ( 𝑑𝑑 ) ; 𝑑𝑑 (0) = 𝑑𝑑 ° System of Linear ODEs 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑑𝑑 = 𝑎𝑎𝑑𝑑 + 𝑏𝑏 ( 𝑑𝑑 ) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 − 𝑎𝑎𝑑𝑑 = 𝑏𝑏 ( 𝑑𝑑 ) Integrating factor 𝑒𝑒 −𝑎𝑎𝑑𝑑 Scalar Case 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 − 𝑎𝑎 ( 𝑑𝑑 ) 𝑑𝑑 = 𝑏𝑏 ( 𝑑𝑑 ) IF 𝑒𝑒 −∫ 𝑎𝑎 ( 𝑑𝑑 ) 𝑑𝑑𝑑𝑑 𝑒𝑒 −𝑎𝑎𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 − 𝑎𝑎𝑑𝑑� → 𝑒𝑒 −𝑎𝑎𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑑𝑑 𝑑𝑑�𝑒𝑒

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