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Different Methods 11-23

# Different Methods 11-23 - 1 1 2 = 1 1 = 1(2 1 = 2 det = 0...

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HW Help 𝐴𝐴 ↔ 𝑘𝑘 1 𝑘𝑘 1 𝐵𝐵 → 𝑘𝑘 2 𝐶𝐶 𝑑𝑑𝐶𝐶 𝐴𝐴 𝑑𝑑𝑑𝑑 = −𝑘𝑘 1 𝐶𝐶 𝐴𝐴 + 𝑘𝑘 1 𝐶𝐶 𝐵𝐵 𝑑𝑑𝐶𝐶 𝐵𝐵 𝑑𝑑𝑑𝑑 = 𝑘𝑘 1 𝐶𝐶 𝐴𝐴 ( 𝑘𝑘 2 + 𝑘𝑘 1 ) 𝐶𝐶 𝐵𝐵 𝑑𝑑𝐶𝐶 𝐶𝐶 𝑑𝑑𝑑𝑑 = 𝑘𝑘 2 𝐶𝐶 𝐵𝐵 det( 𝐴𝐴 − 𝜆𝜆𝜆𝜆 ) = 0 = −𝜆𝜆 𝜆𝜆 ( 𝜆𝜆 2 + 𝑏𝑏𝜆𝜆 + 𝑐𝑐 ) = 0 ← 𝑑𝑑𝑑𝑑 𝑛𝑛𝑑𝑑𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑛𝑛𝑑𝑑 𝑒𝑒 ( 𝑑𝑑 + ∆𝑑𝑑 ) = 𝑒𝑒 ( 𝑑𝑑 ) + ∆𝑑𝑑𝑡𝑡 ( 𝑒𝑒 ( 𝑑𝑑 ), 𝑑𝑑 ) Implicit Euler 𝑒𝑒̇ = 𝑡𝑡 ( 𝑒𝑒 ( 𝑑𝑑 ), 𝑑𝑑 ) Implicit Euler takes the tangent line of a graph calculated at time, t, and extrapolates for the value for x at time, 𝑑𝑑 + ∆𝑑𝑑 𝑆𝑆𝑆𝑆𝑑𝑑𝑒𝑒𝑒𝑒 𝑒𝑒𝑑𝑑 𝑑𝑑 + ∆𝑑𝑑 = 𝑒𝑒̇ ( 𝑑𝑑 + ∆𝑑𝑑 ) = 𝑡𝑡 ( 𝑒𝑒 ( 𝑑𝑑 + ∆𝑑𝑑 ), 𝑑𝑑 + ∆𝑑𝑑 ) 𝑒𝑒 𝑖𝑖𝑖𝑖𝑒𝑒𝑆𝑆𝑖𝑖𝑐𝑐𝑖𝑖𝑑𝑑 = ( 𝑑𝑑 + ∆𝑑𝑑 ) − 𝑒𝑒 ( 𝑑𝑑 ) =

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