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!