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Linear Systems of Differential Equations

CS 130: Mathematical Methods in Computer Science Ordinary Differential Equations Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 18


Linear Systems of Differential Equations

Ordinary Differential Equations

Linear Systems of Differential Equations First-Order Nonhomogeneous Systems


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3 0

X = AX + B where X =



x y



 and A =

−1 −1

2 1

   −8 ,B= 3


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3 0

X = AX + B where X =



x y



 and A =

−1 −1

2 1

   −8 ,B= 3

First, solve for Xc , the general solution of the associated homogeneous system.


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3 0

X = AX + B where X =



x y



 and A =

−1 −1

2 1

   −8 ,B= 3

First, solve for Xc , the general solution of the associated homogeneous system. Next we find a particular solution Xp for the given nonhomogeneous system.


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3 0

X = AX + B where X =



x y



 and A =

−1 −1

2 1

   −8 ,B= 3

First, solve for Xc , the general solution of the associated homogeneous system. Next we find a particular solution Xp for the given nonhomogeneous system.   a1 Note that B is of the form . a2


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3 0

X = AX + B where X =



x y



 and A =

−1 −1

2 1

   −8 ,B= 3

First, solve for Xc , the general solution of the associated homogeneous system. Next we find a particular solution Xp for the given nonhomogeneous system.   a1 Note that B is of the form . a2 Let us suppose that is also of this form.  Xp  a1 That is, let Xp = . We solve for a1 , a2 . a2


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = −x + 2y − 8 y 0 = −x + y + 3 0

X = AX + B where X =



x y



 and A =

−1 −1

2 1

   −8 ,B= 3

First, solve for Xc , the general solution of the associated homogeneous system. Next we find a particular solution Xp for the given nonhomogeneous system.   a1 Note that B is of the form . a2 Let us suppose that is also of this form.  Xp  a1 That is, let Xp = . We solve for a1 , a2 . a2 X = Xc + Xp


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 6x + y + 6t y 0 = 4x + 3y − 10t + 4


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 6x + y + 6t y 0 = 4x + 3y − 10t + 4  Let Xp be of the form

a1 a2



 t+

b1 b2




Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 4x + 13 y − 3et y 0 = 9x + 6y + 10et


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 4x + 13 y − 3et y 0 = 9x + 6y + 10et  Let Xp be of the form

a1 a2



et


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 2x − y + cos 2t y 0 = 3x − 2y


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 2x − y + cos 2t y 0 = 3x − 2y What do you think should be the form of Xp ???


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 3x − 3y + 4 y 0 = 2x − 2y − 1


Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 3x − 3y + 4 y 0 = 2x − 2y − 1 What do you think should be the form of Xp ???


Linear Systems of Differential Equations

Variation of Parameters

Example x0 = 3x − 3y + 4 y 0 = 2x − 2y − 1


Linear Systems of Differential Equations

Variation of Parameters

Example x0 = 3x − 3y + 4 y 0 = 2x − 2y − 1 Let F (t) be a fundamental matrix of the homogeneous system X 0 = AX. A particular solution of the nonhomogeneous system X 0 = AX + B is Z Xp = F F −1 B dt


Linear Systems of Differential Equations

Exercises Find the general solution of the given nonhomogeneous system x X 0 = AX + B where X = and : y     0 2 1 1. A = ,B= et −1 3 −1  2. A =

0 1

−1 0



 ,B=

sec t 0



et


Linear Systems of Differential Equations

Questions? See you next meeting!


Cs130 day18