Learning Guide Learning Guide

6.2 Ordinary Differential Equations • 249> L(0);[(D (5) )(θ)(0) = 1100 (D(3) )(θ)(0),(D (4) )(θ)(0) = 1100 (D(2) )(θ)(0),(D (3) )(θ)(0) = 1100 D(θ)(0), (D(2) )(θ)(0) = 1100 θ(0) − 1 5 ,D(θ)(0) = − 1 θ(0) + 2]10Now generate the Taylor series.> T := taylor(theta(t), t);T := θ(0) + D(θ)(0) t + 1 2 (D(2) )(θ)(0) t 2 + 1 6 (D(3) )(θ)(0)t 3 + 1 24 (D(4) )(θ)(0) t 4 + 1120 (D(5) )(θ)(0) t 5 + O(t 6 )Substitute the derivatives into the series.> subs( op(L(0)), T );θ(0) + (− 11θ(0) + 2) t + (10 200 θ(0) − 1 10 ) t2 +(− 16000 θ(0) + 1300 ) 1t3 + (240000 θ(0) − 112000 ) t4 +1(−12000000 θ(0) + 1600000 ) t5 + O(t 6 )Now, evaluate the series at the initial condition and convert it to apolynomial.> eval( %, ini );100 − 8 t + 2 5 t2 − 175 t3 + 1 13000 t4 −150000 t5 + O(t 6 )> p := convert(%, polynom);p := 100 − 8 t + 2 5 t2 − 175 t3 + 1 13000 t4 −150000 t5