MEC702_Wk_8_Lect_2 - MEC702 Wk 8 Lect 2 Basic Theory of...

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Unformatted text preview: MEC702 Wk 8 Lect 2 Basic Theory of Systems of ODEs First-Order System of ODEs n The system of n rst-order of ODEs is given by y10 = f1 (t, y1 , . . . , yn ) y20 = f2 (t, y1 , . . . , yn ) yn0 = fn (t, y1 , . . . , yn ) .. . .. . .. . This can be written as a vector equation ~y 0 = f~(t, ~y ) 0 f1 y1 y1 f2 y2 y20 where ~y 0 = .. , ~y = .. and f~ = .. . . . . 0 fn yn yn A solution of the system on some interval (a, b) is a set of functions y1 = h1 (t), y2 = h2 (t), ..., yn = hn (t) on (a, b) that satisfy thesystem throughout this interval. Writing the 'solution h1 (t) h2 (t) vector' as ~h(t) = .. , we do so instead as ~y = ~h(t). . ht (t) IVP An Initial Value initial conditions Problem for the system consists of the systems and n given y1 (t0 ) = K1 , y2 (t0 ) = K2 , 1 ..., yn (t0 ) = Kn and as in vector form K1 K2 ~ = ~y (t0 ) = K .. . . Kn Linear System • We call a system a it can be written as linear system if it is linear in y , y , . . . , y 1 2 n, y10 = a11 (t)y1 + a12 (t)y2 + . . . + a1n (t)yn + g1 (t) y20 = a21 (t)y1 + a22 (t)y2 + . . . + a2n (t)yn + g2 (t) .. . .. . .. . yn0 = an1 (t)y1 + an2 (t)y2 + . . . + ann (t)yn + gn (t). that is, if • In vector form, this can be written as a11 a21 where A = .. . an1 a12 a22 .. . an2 ... ... .. . ... ~y 0 = A~y + ~g a1n y1 g1 y2 g2 a2n .. , ~y = .. and ~g = .. . . . . ann yn gn • The linear system is called homogeneous if ~g = ~0; otherwise it is nonhomogeneous if ~g 6= ~0. (Superposition/ Linearity Principle) If ~y and ~y are solutions of the homogeneous linear system ~y = A~y on some interval, then Theorem. 1 ~y = c1 ~y1 + c2 ~y2 is also a solution of the homogeneous system. 2 2 General Solution Let ~y1 , ~y2 . . . , ~yn be a basis of solutions of the homogeneous system ~y 0 = A~y on some interbal I . We call the linear combination ~y = c1 ~y1 + c2 ~y2 + · · · + cn ~yn is a general solution of the system on I . Each matrix ~yk is a matrix of the form y1k y2k ~yk = . , .. ynk hence forming a n × n matrix  Y = ~y1 ~y2 ··· y11  y21 ~yn = . .. yn1 Wronskian The determinant of Y is called the W (~y1 , ~y2 , . . . , ~yn ) y12 y22 ··· ··· yn2 ··· .. . Wronskian of y11 y21 = . .. yn1 3 .. . y1n y2n .. . . ynn ~y1 , ~y2 , . . . , ~yn , written y12 · · · y1n y22 · · · y2n .. .. .. . . . yn2 · · · ynn Constant Coecient System. Phase Plane Method Eigenvalues and Eigenvectors (General Solution of the Homogeneous Linear System) If the constant matrix A in the linear system ~y = A~y has a linearly independent set of n eigenvalues, Theorem. 0 λ1 , λ2 , . . . , λn , with corresponding linearly independent eigenvectors ~x1 , ~x2 , . . . , ~xn . Then the corresponding solutions are ~y = c1 ~x1 eλ1 t + c2 ~x2 eλ2 t + . . . + cn ~xn eλn t . The Wronskian for a constant coecient system is given by y11 y21 W = yn1 ... y12 y22 ··· ··· yn2 ··· x11 y1n x21 y2n = e(λ1 +λ2 +...+λn )t xn1 ynn ... ... . . . ... x12 x22 ··· ··· xn2 ··· x1n x2n . xnn ... . . . ... Graphing Solution in the Phase Plane Consider the two rst-order ODE with constant coecient ~y 0 = A~y ; in components, y10 = a11 y1 + a12 y2 y20 = a21 y1 + a22 y2 .   y1 (t) We can graph solution of ~y (t) = in two ways: y2 (t) • as two curves over the t-axis, one for each component of ~y (t); • as a single curve in the y1 y2 -plane (phase plane). Such a curve as a trajectory. The y1 y2 -plane is called the phase plane. If the phase plane is lled with trajectories of y10 and y20 , we obtain the called phase portrait. Example. Tra jectories in the Phase Plane. Find and graph the solution of the system.  −3 ~y = A~y = 1 4  1 ~y . −3 The eigenvalues are λ1= −2    and λ2 = −4. The corresponding eigenvectors 1 1 are ~x1 = and ~x2 = for the respective eigenvalues. This gives the solution 1 −1       y1 1 −2t 1 ~y = = c1 ~y1 + c2 ~y2 = c1 e + c2 e−4t . y2 1 −1 5 ...
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