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# Class Note for MATH 322 at UA

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Date Created: 02/06/15
Chapters 1 2 4 Ordinary Differential Equations Sections 11 17 22 26 27 42 amp 43 Chapters 124 Ordinary Differential Equations Definitions Existence and uniqueness of solutions 1 Ordinary differential equatin 0 An ordinary differential equation of order n is an equation of the form d d d 1 y y E fltxydx 1 dX 7 dX 7 1 o A solution to this differential equation is an n times differentiable function yX which satisfies 0 Example Consider the differential equation y 2yy0 o What is the order of this equation 0 Are y1x 2 ex and y2x xeX solutions of this differential equation 0 Are y1x and y2x linearly independent Ordinary differential equations Li fferential eq 5 Nonhc neous linear eq Definitions Existence and uniqueness of solutions Initial and boundary conditions 0 An initial condition is the prescription of the values ofy and of its n 1st derivatives at a point X0 dy dnil m0 Yo7 Em yldXTxo mi 2 where yo yl yn1 are given numbers 0 Boundary conditions prescribe the values of linear combinations of y and its derivatives for two different values of X 0 ln MATH 254 you saw various methods to solve ordinary differential equations Recall that initial or boundary conditions should be imposed after the general solution of a differential equation has been found Chapters 124 Ordinary Differential Equations Ordinary d39ff Lir fferential eq Nonhorm neous linear Definitions Existence and uniqueness of solutions bysettingY y77E7 H7W 0 Existence and uniqueness of solutions if F in 3 is continuously differentiable in the rectangle RX7y7 lXX0llta7 ltb7 a7bgt07 then the initial value problem dY EZFOGYL YX0 Y07 has a solution in a neighborhood of X07 Y0 Moreover this solution is unique Chapters 124 Ordinary Differential Equations Definitions Ordinary dif ential equations M u n 5 Existence and uniqueness of solutions 39 differential e lln e eons linear e e t s lt Existence and uniqueness of solutions continued 0 Examples o Does the initial value problem y 23 y 0 y0 1 yO 0 have a solution near x 07 y 17 y 0 If so is it unique 0 Does the initial value problem y W y0 yo have a unique solution for all values of yo 0 Does the initial value problem y y2 y1 1 have a solution near x 1 y 1 Does this solution exist for all values of x Chapters 124 Ordinary Differential Equations Ordina differential e uations 7 W q lt Definitions Existence and uniqueness of solutions Existence and uniqueness for linear systems 0 Consider a linear system of the form dY E AXY Bx7 where Y and BX are n X 1 column vectors and AX is an n X n matrix whose entries may depend on X 0 Existence and uniqueness of solutions If the entries of the matrix AX and of the vector BX are continuous on some open interval I containing X0 then the initial value problem dY AXY BX7 YX0 2 Y0 dX has a unique solution on I Chapters 124 Ordinary Differential Equations Definitions Existence and uniqueness of solutions Ordinary dif ential equations 39 differential e 39 39 quot s lln e eons linear e e t s s Existence and uniqueness for linear systems continued 0 Examples o Apply the above theorem to the initial value problem y 2y y 3x y0 1 yO 0 0 Does the initial value problem y4 X3y 07 W 17 yO 17 y 0 07 y30 0 have a unique solution on the interval 11 Chapters 124 Ordinary Differential Equations C i39 aw differential equations General facts Linear differenti quations and systems Homogeneous linear equatio with constant coefficients hiemnemogeneous i xns and systen39ls Homogeneous linear systems th constant coefficients 3 Linear differential equations and systems a The general solution of a homogeneous linear equation of order n is a linear combination of n linearly independent solutions 0 As a consequence if we have a method to find n linearly independent solutions then we know the general solution 0 ln MATH 254 you saw methods to find linearly independent solutions of homogeneous linear ordinary differential equations with constant coefficients 0 This includes linear equations of the form 2y by cy O and linear systems of the form d A Y where A is an X n X n constant matrix and YX is a column vector in R Chapters 124 Ordinary Differential Equations c General facts Linear differential equ s Homogeneous linear equations with constant coefficients l ilcnl39Icmegeneous linear equ c a an c s s Homogeneous linear systems with constant coefficients Linear differential equations and systems continued 0 A set y1X7y2X7 7ynX of n functions is linearly independent if its Wronskian is different from zero 0 Similarly a set of n vectors Y1X7 Y2X7 7 YnX in R is linearly independent if its Wronskian is different from zero 0 The Wronskian of n functions y1X7 y2X7 7 ynX is given by Y1 Y2 yn Y1 Y2 yn W1727quot 7Yn y1 y2 ynH yln21 y2n21 y 1 Chapters 124 Ordinary Differential Equations Cir aw differential equations General facts Linear differenti quations and systems Homogeneous linear equatio with constant coefficients lcnhomogeneous i mums and systen39ls Homogeneous linear systems th constant coefficients Linear differential equations and systems continued 0 The Wronskian of n vectors Y1X7 Y2X7 7 YnX in R is given by WY17Y27 7YndetY1 Y2 Yn7 where Y1 Y2 Yn denotes the n X n matrix whose columns are Y1X7 Y2X7 7 YnX 0 Finding n linearly independent solutions to a homogeneous linear differential equation or system of order n is equivalent to finding a basis for the set of solutions 0 The next two slides summarize how to find linearly independent solutions in two particular cases Chapters 124 Ordinary Differential Equations General facts Linear differential equ Homogeneous linear equations with constant coefficients l ilcnhomogeneous linear equ t a a 39 c Homogeneous linear systems with constant coefficients Homogeneous linear equations with constant coefficients To find the general solution to an ordinary differential equation of the form 2y bj l cy 0 where 37 b7 C E R proceed as follows 0 Find the characteristic equation 2A2 bA c O and solve for the roots A1 and A2 9 If b2 42c gt 0 then the two roots are real and the general solution is y C1egt 1X CzeAZX 9 If b2 42c lt O the two roots are complex conjugate of one another and the general solution is of the form y eax C1 cos x C2 sin x where a 5Re1 E f and m1 V4 32Ca b 0 If b2 42c 0 then there is a double root A 2 f and the general solution is y C1 C2X e Chapters 124 Ordinary Differential Equations Qi39clinarv differential equations General facts Linear differe quations and systems Homog 0le linear equations with constant coefficients lonhomogeneous i mums and systems Homogeneous linear systems with constant coefficients Homogeneous linear systems with constant coefficients dY To find the general solution of the linear system A Y where A is an n x n matrix with constant coefficients proceed as follows 0 Find the eigenvalues and eigenvectors of A 9 If the matrix has n linearly independent eigenvectors U17 U2 7U associated with the eigenvalues A1xg An then the general solution is Y Z C1U16A1X l C2U26A2X l l CnUneA M where the eigenvalues A may not be distinct from one another and the Ci39s Ai s and Ui39s may be complex If A has real coefficients then the eigenvalues of A are either real or come in complex conjugate pairs If A then the corresponding eigenvectors U and UJ are also complex conjugate of one another Chapters 124 Ordinary Differential Equations C I Cl i nary diffsrem Linear differential equati s Nonhomogeneous linear equations and systems 4 Nonhomogeneous linear equations and systems 0 The general solution y to a nonhomogeneous linear equation of order n is of the form VX WAX ypX7 where yhX is the general solution to the corresponding homogeneous equation and ypX is a particular solution to the nonhomogeneous equation 0 Similarly the general solution Y to a linear system of dY equations E AXY BX is of the form YX YhX YpX7 where YhX is the general solution to the homogeneous system d AXY and YpX is a particular solution to the X no nho mogeneous system Chapters 124 Ordinary Differential Equations Linear differ quot Is Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems continued 0 ln MATH 254 you saw methods to find particular solutions to nonhomogeneous linear equations and systems of equations 0 You should review these methods and make sure you know how to apply them Chapters 124 Ordinary Differential Equations

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