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Problem 7. a) Give a definition of Fundamental Matrix. (t). (See is a same as a Wronskian) Φ b) Non-homogeneous system X' = AX + F(t) where C- column matrix of coefficients: c (t) = (x11... X21. Xn1.. Xin X2n Xnn Complementary solution is: X = (t)c, Particular Solution is: C = Then C1 C2 Cn Xp= (t) ¹(t)F(t)dt. General solution of Nonhomogeneous equation : X'= AX + F(t) Is X = (t)C+ (t)/ ¹(t)F(t)dt (1) 11 Solve the problem: X'= 0 2 X+ 1 e¹ -1 3 -1

Problem 7. a) Give a definition of Fundamental Matrix. (t). (See is a same as a Wronskian) Φ b) Non-homogeneous system X' = AX + F(t) where C- column matrix of coefficients: c (t) = (x11... X21. Xn1.. Xin X2n Xnn Complementary solution is: X = (t)c, Particular Solution is: C = Then C1 C2 Cn Xp= (t) ¹(t)F(t)dt. General solution of Nonhomogeneous equation : X'= AX + F(t) Is X = (t)C+ (t)/ ¹(t)F(t)dt (1) 11 Solve the problem: X'= 0 2 X+ 1 e¹ -1 3 -1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 7. a) Give a definition of Fundamental Matrix. (t). (See is a same as a Wronskian) (t)= X11 X21. where C= column matrix of coefficients: C₁ Xn1. b) Non-homogeneous system X' = AX + F(t) Xin X2n Xnn Particular Solution is: Complementary solution is: X = (t)C, C = C1 C2 T Then Cn Xp= o(t) ¹(t)F(t)dt. General solution of Nonhomogeneous equation: X' = AX + F(t) Is X = (t)C+ (t)/ ¹(t)F(t)dt (1) Solve the problem: X'= 0 2 X+ 1 e¹ -1 3 -1 Hint: first find the eigenvalues and corresponding eigenvectors for matrix. Remember (t) is a matrix of solution vectors of homogeneous equation. Then find ¹(t) Then follow formula (1) find general solution of the given system.
Transcribed Image Text:Problem 7. a) Give a definition of Fundamental Matrix. (t). (See is a same as a Wronskian) (t)= X11 X21. where C= column matrix of coefficients: C₁ Xn1. b) Non-homogeneous system X' = AX + F(t) Xin X2n Xnn Particular Solution is: Complementary solution is: X = (t)C, C = C1 C2 T Then Cn Xp= o(t) ¹(t)F(t)dt. General solution of Nonhomogeneous equation: X' = AX + F(t) Is X = (t)C+ (t)/ ¹(t)F(t)dt (1) Solve the problem: X'= 0 2 X+ 1 e¹ -1 3 -1 Hint: first find the eigenvalues and corresponding eigenvectors for matrix. Remember (t) is a matrix of solution vectors of homogeneous equation. Then find ¹(t) Then follow formula (1) find general solution of the given system.
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