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# Class Note for MATH 254 at UA

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Date Created: 02/06/15
Chapters 124 Ordinary Differential Equations Sections 11 17 22 26 27 42 amp 43 Chapters 124 Ordinary ual Equations 0rd inarv d llal equations Definitions Ewstenre and Unlqu ol solu lions 1 Ordinary differential equations n An ordinary differential equation of order n is an equation of dy the form dnil f X7y7 77 y dX dxn 1 o A solution to this differential equation is an n times differentiable function yX which satisfies dX 1 0 Example Consider the differential equation y 2y y0 o What is the order of this equation 0 Are y1X eX and y2X xeX solutions of this differential equation a Are y1X and y2X linearly independent Chapters 124 Ordinary Dif al Equations Definitions noquot and lllllilquot lollnilioiis 9 and uniqueness of solutions Initial and boundary conditions 0 An initial condition is the prescription of the values of y and of its n 7 1st derivatives at a point X0 n71 d d YX0 in 1 X0 Y17dT1Xo yn717 X 2 where yo y1 ynsl 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 124 Ordinary Differential Equations 2 Existence and uniqueness of solutions 0 Equation 1 may be written as a first order system dY F Y 3 dX X7 l l I d d2 dnil T by setting Y y7d 7d 7 i ani 0 Existence and uniqueness of solutions if F in 3 is continuously differentiable in the rectangle RX7Y7 lxixollta7 llYiYollltb7 a7bgt07 then the initial value problem dY F Y W x i has a solution in a neighborhood of X07 Y0 Moreover this solution is unique YX0 Y07 Chapters 124 Ordinary Differential Equations Noni 39 olsoluuons lefiniii and uniqueness ofsolulions an ar Existence and uniqueness of solutions continued 0 Examples 0 Does the initial value problem y i 2y y 0 y0 17 y 0 0 have a solution near X 07 y 17 y 0 If so is it unique 0 Does the initial value problem y W77 M0 yo have a unique solution for all values of yo 0 Does the initial value problem y yQ y1 1 have a solution near X l y 1 Does this solution exist for all values of X Chapters 24 Ordinarv Differential Equations Existence and uniqueness for linear systems 0 Consider a linear system of the form dY E AXY Bo 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 dY 7 Amy Bo dX i YX0 Y0 has a unique solution on I Chapters 124 Ordinarv Differential Equations 0rde dif ntial equalquot mal aquatic and Nonla squash ant 39 ofsolutions Existence and uniqueness for linear systems continued 0 Examples 9 Apply the above theorem to the initial value problem y i 2y y 3X y0 17 y 0 0 0 Does the initial value problem y X3y 3y 0 y0 1 y 0 1 y 0 0 Wm 0 have a unique solution on the interval 711 24 Ordinary Differential Equations o 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 a In MATH 254 you saw methods to find linearly independent solutions of homogeneous linear ordinary differential equations with constant coefficients a This includes linear equations of the form ay by cy 0 and linear systems of the form A Y where A is an x n x n constant matrix and Yx is a column vector in IR Cliaptz 124 Ordinarv Differential Equations masxv differential IInEar equation mm 39 mswlthconstanl mm 3939 quathmsant mung3 v n r gtwllvcmsLana Linear differential equations and systems continued continued a A set y1x7y2x7 7ynx of n functions is linearly independent if its Wronskian is different from zero The wronskian of n vectors Y1X7 Y2X7 7 Y X in Rn Is given by 0 Similarly a set of n vectors Y1X7 Y2X7 7Ynx in R is linearly independent if its Wronskian is different from zero WY17Y2v vYn det lyl Y2 Ynlv 9 The Wronskian of n functions y1x7 y2x7 7 ynx is where Y1 Y2 Yn denotes the n x n matrix whose given by columns are Y1X7 Y2X7 7 Ynx y1 y2 yn 0 Finding n linearly independent solutions to a homogeneous y1 y2 yn linear differential equation or system of order n is equivalent WW1 y2 H y 7 y1 y2 yn to finding a basis for the set of solutions 7 7 7 n 7 a The next two slides summarize how to find linearly y1 1 y2 1 M10771 independent solutions in two particular cases Chapters 24 Ordinary Differential Equations Chapters 1724 Ordinary Differential Equations linear equations mm Islam c Noni I3939 quathmsant mung3 swmswnlu slant Homogeneous linear equations with constant coefficients To find the general solution to an ordinary differential equation of dy the form 3y 7 by Cy 0 Where a b C E R proceed as foOWS To find the general solution of the linear system A Y where l 7 7 7 X A is an n x n matrix with constant coefficients proceed as follows 0 Find the characteristic equation a2 b c 0 and solve 0 Find the eigenvalues and eigenvectors of A for the rOOtS 1 and 239 9 If the matrix has n linearly independent eigenvectors 9 If b2 7 43c gt 0 then the two roots are real and the general U17 U27 39 H 7U assoc39ated mm the efgenYalues solution is y Clexlx C26le A1 A2 A then the general solution Is A Y C1U1e 1X 2U2e 2X CnUne quotX7 9 If b2 7 4ac lt 0 the two roots are complex conjugate Of one where the eigenvalues A may not be distinct from one another and the general solution is of the form another and the C Ai s and U s may be complex 7 a 39 7 7 7b y i e X C1 COSWX CZS39anDY Where a iReO l Z7 If A has real coefficients then the eigenvalues of A are either real and Squ E 23 or come in complex conjugate pairs If A then the corresponding eigenvectors U and Uj are also complex conjugate 2 7 39 7 A 0 If b 7 4ac 7 0 then there Is a double root A 7 723 and the of one another general solution is y C1 ng e 24 Ordinary Differential Equations Chapl 1724 Ordinary Differential Equations Lquot Nonhomo linearequationsand Nonhomogeneous linear equations and systems Nonhomogeneous linear equations and systems continued a The general solution y to a non homogeneous linear equation of order n is of the form yX yhX yplx where yhX is the general solution to the corresponding homogeneous equation and yPX is a particular solution to the non homogeneous equation a In MATH 254 you saw methods to find particular solutions to non homogeneous linear equations and systems of equations 0 Similarly the general solution y to a linear system of a You should review these methods and make sure you know h t th equations AXY 300 is of the form OW o appy em d 3 00 YhX Yplxll 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 124 Ordinary ual Equations Chapters 124 Ordinary Dif al Equations

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