(2) Classify the critical points of linear systems and almost linear systems.

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1 Review for Exam 3 Three type of prolem: () Solve the firt order homogeneou linear ytem x Ax () Claify the critical point of linear ytem and almot linear ytem (3) Solve the high order linear equation uing Laplace tranform Solve the homogeneou linear ytem: x Ax Step: () Write the linear ytem in the matrix form x Ax for ome matrix A () Finhe eigenvalue of A y olving the characteritic equation A λi (3) Loo at the eigenvalue: Two ditinct real eigenvalue λ, λ : - For each eigenvalue, find one eigenvector y olving (A λi)v - Two independent olution x v e λt, x v e λ t - The general olution i x(t) c x (t) + c x (t) A pair of complex eigenvalue λ p + qi, λ p qi: - Find one complex-valued eigenvector v to λ - The complex-valued olution x(t) ve λt ve pt (co qt + i in qt) - Tae the real part and imaginary part of x(t): x (t) an (t) - General olution i x(t) c x (t) + c x (t) A repeated eigenvalue of multiplicity : - Set v, and calculate v (A λi)v - Two linearly independent olution: x (t) v e λt an (t) (v t + v )e λt - The general olution i x(t) c x (t) + c x (t)

2 Example Solve the linear ytem Solution: x x x x x + x () Write the linear ytem in the matrix form x Ax for ome matrix A x x Ax for the matrix A, ecaue x x x () Finhe eigenvalue of A y olving the characteritic equation A λi Characteritic equation: A λi a λ a a a λ λ λ ( λ)( λ) ( ) λ λ + 5 Then the eigenvalue are λ ± ( ) 4 5 ± 4i ± i (3) Loo at the eigenvalue: We have a pair of complex eigenvalue λ ± i, and A A pair of complex eigenvalue λ + i, λ i: Find one complex-valued eigenvector v to λ For eigenvalue λ + i: a λ a (A λi)v a a λ ( + i) a ( + i) a ( i)a + ( ) Thi correpond to the equation: a + ( i) Thi i equivalent to a + ( i) One olution can e a i and Then v λ + i i The complex-valued olution x(t) ve λt ve pt (co qt + i in qt) i i a i an eigenvector aociateo the eigenvalue x(t) ve pt (co qt + i in qt) i e t (co t + i in t) Tae the real part and imaginary part of x(t): x (t) an (t) e t in t Real part x (t) e t anhe imaginary part x co t (t) General olution i x(t) c x (t) + c x (t) e t in t + i e t co t e t co t + i e t in t e t co t e t in t x(t) c e t in t e t co t e t co t + c e t in t

3 Type and taility of critical point in a ytem ax + y () Linear ytem: c x + Critical point: (,) (unique) Finhe eigenvalue of the matrix A y olving the characteritic equation A λi Loo at the eigenvalue and get the type and taility () Almot linear ytem: f (x, y) g(x, y) f (x, y) Critical point: The olution of equation g(x, y) For each critical point (x, y ): Write the linearized ytem a u fx (x Ju, where J(x, y ), y ) f y (x, y ) g x (x, y ) g y (x, y ) Finhe eigenvalue of J(x, y ) y olving the characteritic equation J λi Loo at the eigenvalue and get the type and taility Eigenvalue Sign Linear ytem Almot linear ytem Ditinct eigenvalue Repeated eigenvalue Complex eigenvalue oth poitive oth negative oppoite ign λ >,λ > improper ource (U) improper ource (U) λ <, λ < improper in (AS) improper in (AS) λ < < λ addle (U) addle (U) poitive λ λ > proper/improper ource (U) negative λ λ < proper/improper in (AS) poitive real part negative real part zero real part proper/improper /piral ource (U) proper/improper /piral in (AS) λ, λ a ± i(a > ) piral ource (U) piral ource (U) λ, λ a ± i(a < ) piral in (AS) piral in (AS) λ, λ ±i center (S) center/ piral point (NA)

4 Example 6 # 3 Finhe linearized ytem for each critical point: y x y Solution: Thi ia an almot linear ytem with f (x, y) y and g(x, y) x y y () Finhe critical point: x y From the firt equation y, we get y Plug in thi reult to the econd equation x y, we, get x That i (x + )(x ) Then we have x an Critical point (, ), (, ) () Linearized ytem of critical point (, ): f x (x, y), f y (x, y), g x (x, y) x, g y (x, y) Jacoian matrix J(, ) f x(, ) f y (, ) g x (, ) g y (, ) a λ a Characteritic equation: J λi a a λ λ ( λ)( λ) λ λ + λ (λ + )(λ ) Eigenvalue λ, λ (Two ditinct real eigenvalue, oppoite ign) Type: Saddle (untale) (3) Linearized ytem at critical point (, ): f x (x, y), f y (x, y), g x (x, y) x, g y (x, y) Jacoian matrix J(, ) f x(, ) f y (, ) g x (, ) g y (, ) a λ a Characteritic equation: J λi a λ λ ( λ)( λ) ( ) λ λ + λ + Eigenvalue λ ± 4 part) a Type: Spiral in (Aymptotically tale) ± 7i (A pair of two complex eigenvalue, negative real

5 Ue Laplace tranform to olve ax (t) + x (t) + cx(t) f (t); x() x, x () x () Laplace tranform of IVP: a (x ) + (x ) + c (x) (f ) (x) X () () Uing the formula to rewrite the tranformed equation (x ) X () x() (x ) X () x() x () (3) Get an algeraic equation of X () of degree Solve for X () (4) Do the invere Laplace tranform of X () to get the olution x(t) (Ue partial fraction) f (t) t t n (n ) F() n! n+ t α (α > ) Γ(a + ) a+ e at a co t + in t + coh t inh t f (t) F() e at t n n! ( a) n+ e at co t e at in t eat t in t 3 e at (in t t co t) a ( a) + ( a) + a [( a) + ] [( a) + ]