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 âˆ’ 10t + 4

Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 6x + y + 6t y 0 = 4x + 3y âˆ’ 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 âˆ’ 3et y 0 = 9x + 6y + 10et

Linear Systems of Differential Equations

Method of Undetermined Coefficients

Example x0 = 4x + 13 y âˆ’ 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!