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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 1 2 4 Ordinary Differential Equations Definitions Ordinary differential equations in Existence and uniqueness of solutions Lin entialeqL 39 N o n h 0 inc 1 Ordinary differential equations 0 An ordinary differential equation of order n is an equation of the form dny dy dn ly dX7 7 dxn 1 1 o A solution to this differential equation is an n times differentiable function yX which satisfies 0 Example Consider the differential equation y 2y y0 o What is the order of this equation a Are y1x ex and y2X xeX solutions of this differential equation a Are y1X and y2X linearly independent Ordinary differential equations Linea rcntial equatior l lonhomoge linear equatior 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 d dn l yX0 Yo d X0 Y1 dxnjlXo Yn 17 2 where yo y1 yn1 are given numbers 0 Boundary conditions prescribe the values of linear combinations of y and its derivatives for two different values of X o In 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 1 2 4 Ordinary Differential Equations Definitions Ordinary differential equations l Existence and uniqueness of solutions Lin entialeqL 39 N o n h 0 inc 2 Existence and uniqueness of solutions 0 Equation 1 may be written as a first order system dY F Y dX x 3 dy d2y dn ly T by setting y y73737 7 an1 0 Existence and uniqueness of solutions if F in 3 is continuously differentiable in the rectangle RX7Y7 lXX0l lt37 ltb7 avbgt07 then the initial value problem dY a FX7 Y7 YX0 Y07 has a solution in a neighborhood of XO Y0 Moreover this solution is unique Chapters 1 2 4 Ordinary Differential Equations Definitions Nomhmm a Existence and uniqueness of solutions UI J39DE Existence and uniqueness of solutions continued 0 Examples 9 Does the initial value problem y 2y y 0 y0 1 y 0 0 have a solution near X 0 y 1 y 0 If so is it unique 0 Does the initial value problem yl 7 y0JO 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 1 2 4 Ordinary Differential Equations Ordinar differential e uations y q Definitions Linea rcntlal equale quotIs 1 I r EXIstence and uniqueness of solutions I Oil Wumoge a Existence and uniqueness for linear systems 0 Consider a linear system of the form dY E AXY BU 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 containing X0 then the initial value problem 3 AxY BX YXo Y0 has a unique solution on I Chapters 1 2 4 Ordinary Differential Equations Definitions Existence and uniqueness of solutions 0 Examples 0 Apply the above theorem to the initial value problem y 2y y 3x y0 1 y 0 0 a Does the initial value problem X3yl 07 y0 1 y 0 1 y 0 0 y30 0 have a unique solution on the interval 11 Chapters 1 2 4 Ordinary Differential Equations Ordinary differential equations General facts Linear differential equations and systems Homogeneous linear equations with constant coefficients Norquothomogeneous linear equations and systems Homogeneous linear systems with constant coefficients 3 Linear differential equations and ms 0 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 In 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 ay l by l cy O Y 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 1 2 4 Ordinary Differential Equations Ordinary differential equations General facts Linear differential equ 39 Homogeneous linear equations with constant coefficients l lonhomogenews linear eqta v 39 2quot s Homogeneous linear systems with constant coefficients Linear differential equations and systems continued 0 A set y1Xy2X ynX of n functions is linearly independent if its Wronskian is different from zero 0 Similarly a set of n vectors Y1X Y2X YnX in R is linearly independent if its Wronskian is different from zero a The Wronskian of n functions y1X y2X ynX is given by Y1 YZ Yn y1 y2 yn Wy17y27 7y y1 y2 yn y1n1 y2n1 ynn1 Chapters 1 2 4 Ordinary Differential Equations Ordinary differential equations General facts Linear differential equat systems Homogeneous linear equations with constant coefficients l lonhomogenews linear equations and systems Homogeneous linear systems with constant coefficients Linear differential equations and systems continued 0 The Wronskian of n vectors Y1X Y2X YnX in R is given by WY17 Y27 7y Y2 Yn7 where Y1 Y2 Yn denotes the n x n matrix whose columns are Y1X Y2X 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 1 2 4 Ordinary Differential Equations Ordinary differential equations General facts Linear differential equati s an Homogeneous linear equations with constant coefficients Norhomogeneous linear equatio Ii an a c us 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 ay l by l cy 0 where a b C 6 IR proceed as follows 9 Find the characteristic equation 3A2 l bA l C O and solve for the roots A1 and A2 9 If b2 4ac gt 0 then the two roots are real and the general solution is y C1egt 1x C2egt 2x 3 If b2 4ac lt O the two roots are complex conjugate of one another and the general solution is of the form y eO X C1 cos6x C2 sin x where 04 qu 3 5 and 5 m1 9 If b2 4ac 0 then there is a double root A 2 2 and the general solution is y C1 l C2X e Chapters 1 2 4 Ordinary Differential Equations Ordinary differential equations General facts Linear differential equ ons and systems Homogeneous linear equations with constant coefficients Norhomogeneous linear equations and syetems Homogeneous linear systems with constant coefficients Homogeneous linear systems with constant coefficients dY To find the general solution of the linear system d A Y where X 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 U1 U2 Un associated with the eigenvalues A1 A2 An then the general solution is Y C1 U1egt 1X l C2 U26A2X l l Cn Uneknx where the eigenvalues A may not be distinct from one another and the C s Ai s and U s may be complex If A has real coefficients then the eigenvalues of A are either real or come in complex conjugate pairs If A X then the corresponding eigenvectors U and are also complex conjugate of one another Chapters 1 2 4 Ordinary Differential Equations Linear dif iere al c I Nonhomogeneous linear equations and sy ms 4 Nonhomogeneous linear equations and systems 0 The general solution y to a non homogeneous linear equation of order n is of the form yX yhX ypX7 where yhX is the general solution to the corresponding homogeneous equation and ypX is a particular solution to the non homogeneous equation 0 Similarly the general solution Y to a linear system of dY equations AXY BX is of the form dX W W rpm where YhX is the general solution to the homogeneous dY system d AXY and YpX is a particular solution to the X non homogeneous system Chapters 1 2 4 Ordinary Differential Equations Ordinary d Linear differential eqt Nonhomogeneous linear equ Nonhomogeneous linear equations and systems conued o In MATH 254 you saw methods to find particular solutions to non homogeneous linear equations and systems of equations 0 You should review these methods and make sure you know how to apply them Chapters 1 2 4 Ordinary Differential Equations

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