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CHAPTER 5 Higher-Order Linear Differntial Equations Second-order DE: Second-order linear DE: Note: A,B,C,F function of x only Second-order homogeneous linear DE: Note: F(x)=0 Second-order homogeneous linear DE (with constant coefficients): Note: A, B, C, are constants (all) (a,c,d,e) (d,e) (d)

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CHAPTER 5 Higher-Order Linear Differntial Equations Second-order homogeneous linear DE: Note: F(x)=0 (d,e) are solutions ?? (verify) Consider the homogeneous 2ed-order linear DE: Let W = the set of all solutions of (*) (*) Let F = the set of all real-valued functions W is a subspace of F dim(W)=2

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CHAPTER 5 Higher-Order Linear Differntial Equations Second-order homogeneous linear DE: Consider the homogeneous 2ed-order linear DE: (*) W is a subspace of F dim(W)=2 Give me other solutions???

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous 2ed-order linear DE: (*) How to solve homog. 2ed-order linear DE: 1 Find two linearly independent solutions for (*) 2 The general solution for (*)

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous 2ed-order linear DE (with constant coeff): (*) How to y1 & y2: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real2 non-real

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous 2ed-order linear DE (with constant coeff): (*) How to y1 & y2: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real2 non-real

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous 2ed-order linear DE (with constant coeff): (*) How to y1 & y2: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real2 non-real

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous nth-order linear DE: (*) How to solve homog. nth-order linear DE: 1 Find n linearly independent solutions for (*) 2 The general solution for (*)

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Consider the homogeneous nth-order linear DE (with constant coeff): How to y1, y2,.. yn: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real non-real

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Consider the homogeneous nth-order linear DE (with constant coeff): How to y1, y2,.. yn: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real non-real

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous 2ed-order linear DE (with constant coeff): (*) How to y1 & y2: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real2 non-real

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Consider the homogeneous nth-order linear DE (with constant coeff): How to y1, y2,.. yn: 1 Find the characteristic equation Find the roots of (**) (**) 2 Distinct realrepeated real non-real

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CHAPTER 5 Higher-Order Linear Differntial Equations Ploynomial Operator Write in operator form Euler’s Formula

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the homogeneous nth-order linear DE: (*) How to solve homog. nth-order linear DE: 1 Find n linearly independent solutions for (*) The general solution for (*) 2 Consider the non-homogeneous nth-order linear DE: (**) How to solve non-homog. nth-order linear DE: 1 Solve the associated homog. DE (*) Find a particular solution for (**) 2 (complementary function) The general solution for (**) 3

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CHAPTER 5 Higher-Order Linear Differntial Equations Consider the non-homogeneous nth-order linear DE: (**) How to solve homog. nth-order linear DE: 1 Solve the associated homog. DE (*) Find a particular solution for (**) 2 (complementary function) The general solution for (**) 3

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