Introduction to Differential Equations J. M. Veal, Ph. D. version 13.08.30 Contents 4 Series Solutions of Linear Equations 4 4.1 Solutions About Ordinary Points . 4 1 Introduction to Differential Equations 2 4.2 Solutions About Singular Points . 4 1.1 Definitions and Terminology . 2 4.3 Special Functions (in Brief) . 5 1.2 Initial-Value Problems . 2 1.3 Differential Equations as Mathematical Models . 2 5 Systems of Linear First-Order Differential Equations 5 5.1 Preliminary Theory . 5 2 First-Order Differential Equations 2 5.2 Homogeneous Linear Systems with Constant Coefficients . 5 2.1 Separable Variables . 2 2.2 Linear Equations . 2 6 The Laplace Transform 5 2.3 Exact Equations . 2 6.1 Definition of the Laplace Transform . 5 2.4 Solutions by Substitutions . 2 6.2 Inverse Transform and Transforms of Derivatives . 6 6.3 Translation Theorems . 6 3 Higher-Order Differential Equations 3 6.4 tn Factors and Convolution . 6 3.1 Preliminary Theory: Linear Equations . 3 6.5 Dirac Delta Function . 6 3.2 Reduction of Order . 3 3.3 Homogeneous Linear Equations with Constant Coefficients . 3 7 Modeling with Differential Equations 7 3.4 Undetermined Coefficients { Annihilator Approach . 3 7.1 Linear Models . 7 3.5 Variation of Parameters . 4 7.2 Nonlinear Models . 7 3.6 Cauchy-Euler Equation . 4 7.3 Modeling with Systems of Differential Equations . 7 A Transforms & Inverse Transforms 8 1 J. 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Veal, Introduction to Differential Equations 2 1 Introduction to Differential Equations • singular points. explanation 1.1 Definitions and Terminology • reciprocal. example • differential equations, type, order, linearity. explanation 2.3 Exact Equations • solution, interval, trivial, explicit/implicit, families, singular. explanation • differential, exact equation. explanation 1.2 Initial-Value Problems • criterion for an exact differential. theorem, proof • existence of a unique solution for first-order. theorem • METHOD { Exact Equation: 1.3 Differential Equations as Mathematical Models BM BN Mpx; yq dx Npx; yq dy 0; with • physics, chemistry, biology, engineering. examples By Bx { integrate M wrto x, take derivative wrto y, set equal to N, find gpyq 2 First-Order Differential Equations 2.1 Separable Variables 2.4 Solutions by Substitutions • METHOD { Separable Equation: • homogeneous function, homogeneous differential equation. explanation dy • METHOD { Homogeneous Equation: gpxq hpyq dx Mpx; yq dx Npx; yq dy 0; { separate variables p q α p q • losing a solution. example with M tx; ty t M x; y α • dissimilar expressions. example and Nptx; tyq t Npx; yq { substitute y ux and dy u dx x du (or substitute x vy and 2.2 Linear Equations dx v dy y dv) • linear, homogeneous, nonhomogeneous, standard form. explanation • METHOD { Bernoulli's Equation: • property, variation of parameters, integrating factor. explanation, deriva- dy tions P pxqy fpxqyn dx • METHOD { Linear Equation: { for n 0; 1, substitute u y1¡n dy a1pxq a0pxqy gpxq dx • METHOD { Equation Can Be Reduced To Separable: { find standard form, multiply by integrating factor, l.h.s. is ³ dy d p q fpAx By Cq; with B 0 re P x dxys, integrate dx dx • constant of integration. discussion { substitute u Ax By C J. 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Veal, Introduction to Differential Equations 3 3 Higher-Order Differential Equations 3.3 Homogeneous Linear Equations with Constant Coefficients 3.1 Preliminary Theory: Linear Equations • second-order with constant coefficients, exponential solution. explanation • auxiliary equation, three cases of roots, Euler's formula. explanation • existence of a unique solution. theorem • METHOD { Homogeneous Linear Second-Order Equation with Constant • boundary value problem. discussion Coefficients: 2 1 a2y a1y a0y 0 • homogeneous, nonhomogeneous. explanation { ay2 by1 cy 0, leads to auxiliary equation am2 bm c 0, • nth-order differential operator, linear operator. explanation three cases for m1 & m2 m x m x { distinct real roots: y c1e 1 c2e 2 , repeated real roots: y m x m x αx • superposition principle { homogeneous equations. theorem, proof, corol- c1e 1 c2xe 1 , conjugate complex roots: y e pc1 cos βx laries c2 sin βxq • linear dependence/independence, Wronskian. definitions • METHOD { Homogeneous Linear Higher-Order Equation with Constant Coefficients: pnq pn¡1q 2 1 • criterion for linearly independent solutions. theorem any an¡1y ¤ ¤ ¤ a2y a1y a0y 0 • fundamental set of solutions. definition { find at least one root of auxiliary equation via algebraic guess { long division, guess again, repeat until arrive at second-order • general solution { homogeneous equations. theorem, proof • multiplicity. explanation 3.4 Undetermined Coefficients { Annihilator Approach Exam 1 covers material up to here. • nth-order differential operator, linear operator. review • factoring operators, annihilator operator. explanation 3.2 Reduction of Order • METHOD { Nonhomogeneous Linear Second-Order or Higher-Order Equation with Constant Coefficients: • linear second-order homogeneous equation, standard form. explanation pnq pn¡1q 2 1 any an¡1y ¤ ¤ ¤ a2y a1y a0y gpxq • general case, formula, check. derivation { find complementary function, yc [i.e., solve corresponding homoge- neous equation Lpyq 0] • METHOD { Homogeneous Linear Second-Order Equation { Second Solu- p ¡ qn n¡1 αx tion: { D α annihilates x e ¡ 2 1 { rD2 ¡ 2αD pα2 β2qsn annihilates xn 1eαx cos βx and/or a2pxqy a1pxqy a0pxqy 0 xn¡1eαx sin βx 1 p q p q { find standard form, substitute y upxqy1pxq and corresponding y { operate with L1 on both sides of L y g x 2 1 and y , substitute w u { find general solution, y, to L1Lpyq 0 » ³ ¡ p q ¡ p q e P x dx { given yp y yc, solve for coefficients in yp by substituting yp into { y2 y1 x 2p q dx yx x Lpyq gpxq J. 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Veal, Introduction to Differential Equations 4 3.5 Variation of Parameters 4 Series Solutions of Linear Equations • assumptions. explanation 4.1 Solutions About Ordinary Points • METHOD { Nonhomogeneous Linear Second-Order Equation: • power series. review 2 1 a2pxqy a1pxqy a0pxqy gpxq • ordinary point, singular point. definition { find standard form, find yc (must be given yc if nonconstant coeffi- cients), substitute yp u1pxqy1pxq u2pxqy2pxq and corresponding • existence of power series solution. theorem 1 2 1 1 y and y , assume u1 & u2 are such that u1y1 u2y2 0, arrive at 1 1 1 1 p q u1y1 u2y2 f x • METHOD { Homogeneous Linear Second-Order Equation About an Or- { Cramer's rule gives u1 W {W and u1 W {W , where W (the dinary Point: 1 1 2 2 2 1 Wronskian), W , and W are given by a2pxqy a1pxqy a0pxqy 0 § 1 § 2 § § § § § y y § § 0 y § § y 0 § § 1 2 § § 2 § § 1 § { If P pxq and Qpxq of standard form are analytic at x0, then x0 is an W § 1 1 § W1 § p q 1 § W2 § 1 p q § y1 y2 f x y2 y1 f x ordinary point ° { integrate to find u & u and hence y and then have y y y 8 p ¡ qn 1 2 1 2 p c p { substitute y n0 cn x x0 and corresponding y and y • higher-order equations. explanation { add all series: ensure powers of x are in phase and ensure summation indices start with same number ° 3.6 Cauchy-Euler Equation 8 p ¡ qk { use identity property (if k0 ck x a 0, then ck 0 for all k) • second-order, xm solution. explanation to establish recurrence relation, find ck for increasing k until patterns are recognizable • auxiliary equation, three cases of roots, note before using Euler's formula. { group all c terms and c terms to find y c y pxq c y pxq explanation 0 1 0 1 1 2 { for three-term recurrence relation, first let c1 0 to find y1pxq and • METHOD { Homogeneous Second-Order Cauchy-Euler Equation: second let c0 0 to find y2pxq ax2y2 bxy1 cy 0 { ax2y2 bxy1 cy 0, leads to auxiliary equation am2 pb¡aqm c 4.2 Solutions About Singular Points 0, three cases for m1 & m2 • regular and irregular singular points. definition m m { distinct real roots: y c1x 1 c2x 2 , repeated real roots: y m m α c1x 1 c2x 1 ln x, conjugate complex roots: y x rc1 cos pβ ln xq • Frobenius' Theorem. c2 sin pβ ln xqs • indicial equation, indicial roots. explanation • reduction to constant coefficients. explanation • METHOD { Homogeneous Linear Second-Order Equation About a Regu- • METHOD (alternate): lar Singular Point (Method of Frobenius): { substitute x et, solve, resubstitute t ln x. 2 1 a2pxqy a1pxqy a0pxqy 0 2 Exam 2 covers material between Exam 1 and here. { If ppxq px ¡ x0qP pxq and qpxq px ¡ x0q Qpxq (where P & Q are from standard form) are analytic at x0, then x0 is an regular singular point J. 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Veal, Introduction to Differential Equations 5 { substitute • general solution { homogeneous systems. theorem ¸8 ¸8 r n n r • general solution { nonhomogeneous systems. theorem y px ¡ x0q cnpx ¡ x0q cnpx ¡ x0q n0 n0 5.2 Homogeneous Linear Systems with Constant Coefficients and corresponding y1 and y2 • eigenvalue, eigenvector. explanation { add all series: ensure powers of x are in phase and ensure summation indices start with same number • general solution { homogeneous systems. theorem, example { equate to 0 the total coefficient of the lowest power of x, solve (indicial • Gauss-Jordan elimination. explanation equation) for roots (two values of r) ° 8 p ¡ qk • METHOD { Homogeneous Linear First-Order System: { use identity property (if k0 ck x a 0, then ck 0 for all k) to ¤ ¤ ¤ establish two recurrence relations using (one for each root of indicial x1ptq a11ptq a12ptq ¤ ¤ ¤ a1nptq x1ptq ¦ ¦ ¦ equation), find ck for increasing k until patterns are recognizable ¦ x2ptq ¦ a21ptq a22ptq ¤ ¤ ¤ a2nptq ¦ x2ptq d p q p q ¦ . ¦ . ¦ . { omit c0 since y C1y1 x C2y2 x dt ¥ . ¥ . ¥ . xnptq an1ptq an2ptq ¤ ¤ ¤ annptq xnptq 4.3 Special Functions (in Brief) (or, equivalently, X1 AX) • Bessel's equation of order ν.