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SM212 Lecture 15 Spring 2006

Assignments. Due:Problems, p. 167,

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SM212 Lecture 15 Spring 2006

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    1. SM212 Lecture 15 Spring 2006 §4 Linear 2nd/Higher LHODE’s §4.2b/§6.2 Constant Coefficient, Distinct and Repeated roots

    2. Assignments Due: Problems, p. 167, #1, #9, #13, #21 Due: Friday, 3 February 2006 Pending: Problems, p. 167, #4, #11; p. 331, #3, #4 Due: Monday, 6 February 2006 Assign: Problems, p. 177, #1, #5, #23, #28EC Due:Tuesday, 7 February 2006

    3. Points to Control (cont) B. Roots for the auxiliary equation: three cases *Real, distinct roots §4.3/6.2 Eigenfunctions *Real roots, some multiple (repeated) §4.3/6.2 Eigenfunctions Complex roots §4.3 Eigenfunctions

    4. Points to Control A. Higher-order linear, homogeneous ODE with constant coefficients Building a fundamental set of solutions Auxiliary equation Roots of the auxiliary equation (eigenvalues) Eigenfunction for each root *Fundamental set of solutions (Wronskian) General solution for the constant coefficient ODE 

    5. Auxiliary equation, eigenvalues Solve the auxiliary equation (roots of the polynomial) for Substitute roots into trial solution y = emx Magic: solns for

    6. Have we got all solutions? Thm 1, p. 162 (exist/unique) 2nd-order LHODE with CCeq (a ?0), then ? a unique soln y(x) to IVP: (interval of definition: ) Defn 1, Thm 2, p. 162. If y1(x), y2(x) a) solve the LHODE (2nd order), Cceq b) y2(x) ?(const)y1(x) (e.g. y1, y2 are lin. indep.) Then solves the IVP for determined from the IC. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE.

    7. Initial value problem the first alternative Example (IVP): ODE+IC: Seek THE soln to IVP: Fund. set of solns: General soln: Impose IC: Soln to IVP:

    8. Auxiliary equation roots: the three alternatives Idea: roots for

    9. Solution of LH-CC-ODE repeated roots case Example (like p. 165) : Auxiliary eq: (try ) One soln: How get 2nd soln? (observe: b2 – 4ac = 0)

    10. Solution of LH-CC-ODE repeated roots case Ans: p. 166 (reduction of order): try MAGIC! 2nd soln: (fundamental set of solns for ODE; general soln) Lucky? No: §6.2. (higher order mult.) Higher order multiplicity: (m-5)^3 = 0 Fund. Set of solns {y1(x) = exp(5x), y2(x) = x exp(5x), y3(x) = x^2 exp(5x)}Higher order multiplicity: (m-5)^3 = 0 Fund. Set of solns {y1(x) = exp(5x), y2(x) = x exp(5x), y3(x) = x^2 exp(5x)}

    11. Have we got all solutions? Thm 1, p. 162 (exist/unique) 2nd-order LHODE with CCeq ( ), then a unique soln y(x) to IVP: (interval of definition: ) Defn 1, Thm 2, p. 162. If a) solve the LHODE (2nd order), Cceq b) If (e.g. y1, y2 are lin. indep.) Then solves the IVP For determined from the IC. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE.

    12. §6.2 Higher-Order Linear homogeneous ODE’s (CC) Pattern recognition: CCEq vs VCEq Idea: algebraic structure of the linear op: Basis for “trying” y(x) = exp(mx) D(exp(mx)) = m exp(mx) (only fn with that property: “eigenfunction”) Basis for “trying” y(x) = exp(mx) D(exp(mx)) = m exp(mx) (only fn with that property: “eigenfunction”)

    13. Higher-order equations: Fundamental Thm of Algebra Context: Fundamental Theorem of Algebra: Example: For P(x) = x3 – 3x2 – 4x + 12 P(2) = 0. So, P(x) = (x – 2) g(x), g(x) = P(x)/(x – 2) = x2 – 2x – 3 =(x – 3)(x + 1)For P(x) = x3 – 3x2 – 4x + 12 P(2) = 0. So, P(x) = (x – 2) g(x), g(x) = P(x)/(x – 2) = x2 – 2x – 3 =(x – 3)(x + 1)

    14. Higher-order equations: Generating solutions Solutions: Fundamental set of solutions? Solve ODE, # = order of ODE Linearly independent? For P(x) = x3 – 3x2 – 4x + 12 P(2) = 0. So, P(x) = (x – 2) g(x), g(x) = P(x)/(x – 2) = x2 – 2x – 3 =(x – 3)(x + 1)For P(x) = x3 – 3x2 – 4x + 12 P(2) = 0. So, P(x) = (x – 2) g(x), g(x) = P(x)/(x – 2) = x2 – 2x – 3 =(x – 3)(x + 1)

    15. Have we got all solutions? How know functions are linearly independent? Defn 1, p. 320 (Wronskian) Given Thm 2, p. 320. If Then are Linearly Independent Fund set of solns: {y1(x), y2(x), y3(x)} Each solve 3rd order LHODE W(y1,y2,y3) ? 0. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE.

    16. Have we got all solutions? (cont) Thm 1, p. 162 (exist/unique) 2nd-order LHODE with CCeq (a ?0), then ? a unique soln y(x) to IVP: (interval of definition: ) Defn 1, Thm 2, p. 162. If y1(x), y2(x) a) solve the LHODE (2nd order), Cceq b) y2(x) ?(const)y1(x) (e.g. y1, y2 are lin. indep.) Then solves the IVP For determined from the IC. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE.

    17. Fund. Solns, LH-ODE

    19. Have we got all solutions? (cont) How know functions are linearly independent? Lemma 1, p. 162 (Wronskian) Compute (p. 164). If Then Fund set of solns: Each solve 2nd order LHODE General soln to ODE: Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE. Have we got them all? Thm 1. Observe the interval of definition is all time General Solution: family of all solutions to ODE: ODE w/o IC ? ODE with IC arbitrary. IC arbitrary constants => c1, c2 arbitrary constants. So 2-parameter family of functions captures all solutions to the ODE.

    20. Auxiliary equation roots: the 2nd alternative Situation B: Auxiliary eq: try y(x) = emx, real, repeated roots Fundamental set of solns; general solution to ODE Quadratic formula for Ax^2 + Bx + C Discriminant: B^2 – 4AC: > 0 (real, distinct); = 0, (real, repeated), < 0 (complex)Quadratic formula for Ax^2 + Bx + C Discriminant: B^2 – 4AC: > 0 (real, distinct); = 0, (real, repeated), < 0 (complex)

    21. Auxiliary equation roots: the 2nd alternative Fundamental set of solns; general solution to ODE, p. 164 (real, repeated roots) IVP: use initial conditions (IC) to evaluate c1 and c2 Same procedure as in first alternative Quadratic formula for Ax^2 + Bx + C Discriminant: B^2 – 4AC: > 0 (real, distinct); = 0, (real, repeated), < 0 (complex)Quadratic formula for Ax^2 + Bx + C Discriminant: B^2 – 4AC: > 0 (real, distinct); = 0, (real, repeated), < 0 (complex)

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