# 29 - Systems of ODEs - Systems of ODEs Higher-order ODEs M...

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Systems of ODEs Higher-order ODEs M ATLAB solvers Systems of Ordinary Differential Equations Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Systems of Ordinary Differential Equations MATH 2070U 1 / 19 Systems of ODEs Higher-order ODEs M ATLAB solvers Systems of Ordinary Differential Equations 1 Systems of ordinary differential equations 2 Higher-order ODEs and 1 st -order systems 3 M ATLAB solvers and 1 st -order IVPs c D. Aruliah (UOIT) Systems of Ordinary Differential Equations MATH 2070U 2 / 19 Systems of ODEs Higher-order ODEs M ATLAB solvers ODEs and IVPs IVPs considered so far all 1 st -order y = f ( t , y ) y ( t 0 ) = y 0 Many ODEs of higher-order, e.g., mass-spring system mx + kx = 0 2 nd -order ODE with 2 ICs x ( 0 ) = X 0 , x ( 0 ) = V 0 Also, systems of ODEs in several dependent variables possible Generic IVP solvers solve 1 st -order IVP systems c D. Aruliah (UOIT) Systems of Ordinary Differential Equations MATH 2070U 4 / 19 Systems of ODEs Higher-order ODEs M ATLAB solvers Coupled springs Coupled mass-spring system: masses m 1 & m 2 , springs k 1 & k 2 Dependent variables x 1 ( t ) and x 2 ( t ) Newton’s laws imply 2 ODEs of 2 nd -order m 1 x 1 = - k 1 x 1 + k 2 ( x 2 - x 1 ) , x 1 ( 0 ) = 0, x 1 ( 0 ) = 1 m 2 x 2 = - k 2 ( x 2 - x 1 ) , x 2 ( 0 ) = 1, x 2 ( 0 ) = 0. Fix parameters, say, m 1 = m 2 = 1, k 1 = 3, k 2 = 2 c D. Aruliah (UOIT) Systems of Ordinary Differential Equations MATH 2070U 5 / 19
Systems of ODEs Higher-order ODEs M ATLAB solvers Two-body problem Two masses m and M interacting by Newtonian gravitation Set origin of coordinate system on larger mass Scale units so GM = 1 Dependent variables x 1 = x 1 ( t ) , x 2 = x 2 ( t ) x 1 = - x 1 / ( x 1 ) 2 + ( x 2 ) 2 3/2 , x 1 ( 0 ) = X 1 , x 1 ( 0 ) = V 1 , x 2 = - x 2 / ( x 1 ) 2 + ( x 2 ) 2 3/2 , x 2 ( 0 ) = X 2 , x 2 ( 0 ) = V 2 c D. Aruliah (UOIT) Systems of Ordinary Differential Equations MATH 2070U 6 / 19 Systems of ODEs Higher-order ODEs M ATLAB solvers 1 st -order IVP system 1 st -order system of ODEs can be written as y = f ( t , y ) Independent variable t , dependent variables y = y 1 y 2 .