Introduction to Differential Equations

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1 Introduction to Differential Equations J. M. Veal, Ph. D. version Contents 1 Introduction to Differential Equations Definitions and Terminology Initial-Value Problems Differential Equations as Mathematical Models First-Order Differential Equations Separable Variables Linear Equations Exact Equations Solutions by Substitutions Higher-Order Differential Equations Preliminary Theory: Linear Equations Reduction of Order Homogeneous Linear Equations with Constant Coefficients Undetermined Coefficients Annihilator Approach Variation of Parameters Cauchy-Euler Equation Series Solutions of Linear Equations Solutions About Ordinary Points Solutions About Singular Points Special Functions (in Brief) Systems of Linear First-Order Differential Equations Preliminary Theory Homogeneous Linear Systems with Constant Coefficients The Laplace Transform Definition of the Laplace Transform Inverse Transform and Transforms of Derivatives Translation Theorems t n Factors and Convolution Dirac Delta Function Modeling with Differential Equations Linear Models Nonlinear Models Modeling with Systems of Differential Equations A Transforms & Inverse Transforms 8

2 J. M. Veal, Introduction to Differential Equations 2 1 Introduction to Differential Equations 1.1 Definitions and Terminology differential equations, type, order, linearity. explanation solution, interval, trivial, explicit/implicit, families, singular. explanation 1.2 Initial-Value Problems existence of a unique solution for first-order. theorem 1.3 Differential Equations as Mathematical Models physics, chemistry, biology, engineering. examples 2 First-Order Differential Equations 2.1 Separable Variables METHOD Separable Equation: separate variables losing a solution. example dissimilar expressions. example 2.2 Linear Equations dy gpxq hpyq dx linear, homogeneous, nonhomogeneous, standard form. explanation property, variation of parameters, integrating factor. explanation, derivations METHOD Linear Equation: a 1 pxq dy dx a 0 pxqy gpxq find standard form, multiply by integrating factor, l.h.s. is d dx re³ P pxqdx ys, integrate constant of integration. discussion singular points. explanation reciprocal. example 2.3 Exact Equations differential, exact equation. explanation criterion for an exact differential. theorem, proof METHOD Exact Equation: Mpx, yq dx Npx, yq dy 0, with BM By BN Bx integrate M wrto x, take derivative wrto y, set equal to N, find gpyq 2.4 Solutions by Substitutions homogeneous function, homogeneous differential equation. explanation METHOD Homogeneous Equation: Mpx, yq dx Npx, yq dy 0, with Mptx, tyq t α Mpx, yq and Nptx, tyq t α Npx, yq substitute y ux and dy u dx x du (or substitute x vy and dx v dy y dv) METHOD Bernoulli s Equation: dy dx for n 0, 1, substitute u y 1 n P pxqy fpxqy n METHOD Equation Can Be Reduced To Separable: substitute u Ax By C dy fpax By Cq, with B 0 dx

3 J. M. Veal, Introduction to Differential Equations 3 3 Higher-Order Differential Equations 3.1 Preliminary Theory: Linear Equations existence of a unique solution. theorem boundary value problem. discussion homogeneous, nonhomogeneous. explanation n th -order differential operator, linear operator. explanation superposition principle homogeneous equations. theorem, proof, corollaries linear dependence/independence, Wronskian. definitions criterion for linearly independent solutions. theorem fundamental set of solutions. definition general solution homogeneous equations. theorem, proof Exam 1 covers material up to here. 3.3 Homogeneous Linear Equations with Constant Coefficients second-order with constant coefficients, exponential solution. explanation auxiliary equation, three cases of roots, Euler s formula. explanation METHOD Homogeneous Linear Second-Order Equation with Constant Coefficients: a 2 y 2 a 1 y 1 a 0 y 0 ay 2 by 1 cy 0, leads to auxiliary equation am 2 bm c 0, three cases for m 1 & m 2 distinct real roots: y c 1 e m 1x c 2 e m2x, repeated real roots: y c 1 e m 1x c 2 xe m1x, conjugate complex roots: y e αx pc 1 cos βx c 2 sin βxq METHOD Homogeneous Linear Higher-Order Equation with Constant Coefficients: a n y pnq a n 1 y pn 1q a 2 y 2 a 1 y 1 a 0 y 0 find at least one root of auxiliary equation via algebraic guess long division, guess again, repeat until arrive at second-order multiplicity. explanation 3.4 Undetermined Coefficients Annihilator Approach n th -order differential operator, linear operator. review factoring operators, annihilator operator. explanation 3.2 Reduction of Order linear second-order homogeneous equation, standard form. explanation general case, formula, check. derivation METHOD Homogeneous Linear Second-Order Equation Second Solution: a 2 pxqy 2 a 1 pxqy 1 a 0 pxqy 0 find standard form, substitute y upxqy 1 pxq and corresponding y 1 and y 2, substitute w u 1» y 2 y 1 pxq dx e ³ P pxqdx y 2 x pxq METHOD Nonhomogeneous Linear Second-Order or Higher-Order Equation with Constant Coefficients: a n y pnq a n 1 y pn 1q a 2 y 2 a 1 y 1 a 0 y gpxq find complementary function, y c [i.e., solve corresponding homogeneous equation Lpyq 0] pd αq n annihilates x n 1 e αx rd 2 2αD pα 2 β 2 qs n annihilates x n 1 e αx cos βx and/or x n 1 e αx sin βx operate with L 1 on both sides of Lpyq gpxq find general solution, y, to L 1 Lpyq 0 given y p y y c, solve for coefficients in y p by substituting y p into Lpyq gpxq

4 J. M. Veal, Introduction to Differential Equations Variation of Parameters assumptions. explanation METHOD Nonhomogeneous Linear Second-Order Equation: a 2 pxqy 2 a 1 pxqy 1 a 0 pxqy gpxq find standard form, find y c (must be given y c if nonconstant coefficients), substitute y p u 1 pxqy 1 pxq u 2 pxqy 2 pxq and corresponding y 1 and y 2, assume u 1 & u 2 are such that u 1 1 y 1 u 1 2 y 2 0, arrive at u1 2 y1 2 fpxq u 1 1 y1 1 Cramer s rule gives u 1 1 W 1{W and u 1 2 W 2{W, where W (the Wronskian), W 1, and W 2 are given by W y 1 y 2 y1 1 y2 1 W 1 0 y 2 fpxq y2 1 W 2 y 1 0 y1 1 fpxq integrate to find u 1 & u 2 and hence y p and then have y y c y p higher-order equations. explanation 3.6 Cauchy-Euler Equation second-order, x m solution. explanation auxiliary equation, three cases of roots, note before using Euler s formula. explanation METHOD Homogeneous Second-Order Cauchy-Euler Equation: ax 2 y 2 bxy 1 cy 0 ax 2 y 2 bxy 1 cy 0, leads to auxiliary equation am 2 pb aqm c 0, three cases for m 1 & m 2 distinct real roots: y c 1 x m 1 c 2 x m 2, repeated real roots: y c 1 x m 1 c 2 x m 1 ln x, conjugate complex roots: y x α rc 1 cos pβ ln xq c 2 sin pβ ln xqs reduction to constant coefficients. explanation METHOD (alternate): substitute x e t, solve, resubstitute t ln x. Exam 2 covers material between Exam 1 and here. 4 Series Solutions of Linear Equations 4.1 Solutions About Ordinary Points power series. review ordinary point, singular point. definition existence of power series solution. theorem METHOD Homogeneous Linear Second-Order Equation About an Ordinary Point: a 2 pxqy 2 a 1 pxqy 1 a 0 pxqy 0 If P pxq and Qpxq of standard form are analytic at x 0, then x 0 is an ordinary point substitute y 8 n0 c npx x 0 q n and corresponding y 1 and y 2 add all series: ensure powers of x are in phase and ensure summation indices start with same number use identity property (if 8 k0 c kpx aq k 0, then c k 0 for all k) to establish recurrence relation, find c k for increasing k until patterns are recognizable group all c 0 terms and c 1 terms to find y c 0 y 1 pxq c 1 y 2 pxq for three-term recurrence relation, first let c 1 0 to find y 1 pxq and second let c 0 0 to find y 2 pxq 4.2 Solutions About Singular Points regular and irregular singular points. definition Frobenius Theorem. indicial equation, indicial roots. explanation METHOD Homogeneous Linear Second-Order Equation About a Regular Singular Point (Method of Frobenius): a 2 pxqy 2 a 1 pxqy 1 a 0 pxqy 0 If ppxq px x 0 qp pxq and qpxq px x 0 q 2 Qpxq (where P & Q are from standard form) are analytic at x 0, then x 0 is an regular singular point

5 J. M. Veal, Introduction to Differential Equations 5 substitute y px x 0 q r 8 c n px x 0 q n n0 and corresponding y 1 and y 2 8 c n px x 0 q n r add all series: ensure powers of x are in phase and ensure summation indices start with same number equate to 0 the total coefficient of the lowest power of x, solve (indicial equation) for roots (two values of r) use identity property (if 8 k0 c kpx aq k 0, then c k 0 for all k) to establish two recurrence relations using (one for each root of indicial equation), find c k for increasing k until patterns are recognizable omit c 0 since y C 1 y 1 pxq C 2 y 2 pxq 4.3 Special Functions (in Brief) Bessel s equation of order ν. explanation Bessel functions, four kinds: first, second, modified first, modified second Legendre s equation of order n. explanation Legendre polynomials 5 Systems of Linear First-Order Differential Equations 5.1 Preliminary Theory linear system. explanation solution vector. definition existence of a unique solution. theorem superposition principle. theorem linear dependence/independence. definition n0 criterion for linearly independent solutions. theorem fundamental set of solutions. definition general solution homogeneous systems. theorem general solution nonhomogeneous systems. theorem 5.2 Homogeneous Linear Systems with Constant Coefficients eigenvalue, eigenvector. explanation general solution homogeneous systems. theorem, example Gauss-Jordan elimination. explanation METHOD Homogeneous Linear First-Order System: x 1 ptq a 11 ptq a 12 ptq a 1n ptq x 2 ptq a 21 ptq a 22 ptq a 2n ptq d dt. x n ptq (or, equivalently, X 1 AX) assume X Ke λt.. a n1 ptq a n2 ptq a nn ptq x 1 ptq x 2 ptq. x n ptq then pa λiqk 0, so non-trivial solution implies det pa λiq 0; must find λ n s (eigenvalues) for each λ n, find K n (corresponding eigenvector) X c 1 K 1 e λ 1t c 2 K 2 e λ 2t distinct real eigenvalues. example c n K n e λnt repeated eigenvalues, multiplicity. example Exam 3 covers material between Exam 2 and here. 6 The Laplace Transform 6.1 Definition of the Laplace Transform integral transform. explanation Laplace transform. definition L tfptqu» 8 0 e st fptqdt

6 J. M. Veal, Introduction to Differential Equations 6 linearity. review exponential order. definition sufficient conditions for existence. theorem, proof 6.2 Inverse Transform and Transforms of Derivatives inverse Laplace transform. explanation linearity, cover-up method. review transform of a derivative. theorem L tf pnq ptqu s n F psq s n 1 fp0q s n 2 f 1 p0q f pn 1q p0q solving equations. explanation 6.3 Translation Theorems first translation theorem. theorem, proof, explanation partial fractions: repeated linear factors. review unit step function. definition, explanation general piecewise-defined function. example second translation theorem. theorem, proof, explanation second translation theorem alternate form. derivation METHOD Homogeneous Linear N th -Order Equation with Constant Coefficients: d n y d n 1 y a n dt n a n 1 dt n 1 a 0 y gptq find L tlpyqu L tgptqu and use linearity; may need L te at fptqu F ps aq " 0, 0 t a U pt aq 1, a t L tfpt aqu pt aqu e as F psq L tgptqu pt aqu e as L tgpt aqu solve algebraic equation for Y psq find yptq L 1 ty psqu 6.4 t n Factors and Convolution multiplying by t n. derivation derivatives of transforms. theorem convolution. definition L tt n fptqu p 1q n dn F psq ds n f g» t commutation of convolution. proof convolution. theorem, proof 0 fpτ qgpt τ qdτ L tf gu L tfptqul tgptqu F psqgpsq inverse of convolution. explanation transform of an integral. explanation 6.5 Dirac Delta Function unit impulse. explanation Dirac delta function. definition δpt t 0 q» 8 transform of delta function. proof sifting property. definition» " 8, t t0 0, t t 0. δpt t 0 qdt 1 fptqδpt t 0 qdt fpt 0 q Exam 4 covers material between Exam 3 and here. The final exam is cumulative up to this point.

7 J. M. Veal, Introduction to Differential Equations 7 7 Modeling with Differential Equations 7.1 Linear Models examples 7.2 Nonlinear Models examples 7.3 Modeling with Systems of Differential Equations examples

8 J. M. Veal, Introduction to Differential Equations 8 A Transforms & Inverse Transforms Transforms of Some Basic Functions: L t1u 1 s L tt n u n!, n 1, 2, 3,... sn 1 L te at u 1 s a L tsin ktu k s 2 k 2 L tcos ktu s s 2 k 2 L tsinh ktu k s 2 k 2 L tcosh ktu s s 2 k 2 Some Inverse Transforms: " * 1 1 L 1 s " * n! t n L 1 s n 1, n 1, 2, 3,... " * 1 e at L 1 s a " * sin kt L 1 k s 2 k 2 " * cos kt L 1 s s 2 k 2 " * sinh kt L 1 k s 2 k 2 " * cosh kt L 1 s s 2 k 2

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