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# DIFFERENTIAL EQUATIONS MATH 308

Texas A&M

GPA 3.6

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This 5 page Study Guide was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Study Guide belongs to MATH 308 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/226054/math-308-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 3 0 8 Review Final Exam Dec 3 2007 1 Know the following de nitions and theorems rmmrnpnsrs FF39QPFE F ltrgtw order of a differential equation linear differential equation I I A existence uniqueness theorem for first order differential equations exact and separable rst order equations phase line for an autonomous first order differential equation Wronskian linearly independent functions on an interval characteristic equation of a second order linear constant coef cient differential equation phase plane for a system of two rst order autonomous differential equations in two unknown functions definition of linearity for a system of differential equations eigenvalues and eigenvectors linearly independent vectors linearly independent set of vector valued functions on an interval 61 b Wronskian of a set of vector valued functions fundamental solution set existenceuniqueness theorem for a first order system of differential equations equilibrium points Laplace transform convolution of two functions convolution theorem properties of the Dirac delta function singular and regular singular points of a second order differential equation You should be able to do the following a b c solve mixing problems and population growth problems nd exact solutions to rst order linear and separable differential equations nd exact solutions to second order linear constant coef cient differential equations know how to solve CauchyEuler second order differential equations reduction of order technique which is used to nd a second solution to a DE variation of parameters which is used to nd a particular solution to a nonhomogeneous DE sketch a few lines in a phase plane nd eigenvalues and eigenvectors of 2 x 2 matrices construct a fundamental solution set for linear constant coef cient systems of differential equations and the general solution to such a system solve an initial value problem for a constant coef cient rst order system be able to use the Laplace transform to convert a differential equation into an algebraic equation use the table of Laplace transforms to compute the inverse Laplace transform of a function use the definition of Laplace transform to verify any of the formulas in the table of transforms compute the convolution of two functions use the delta function to model an impulse force nd power series solution of a second order differential equation around an ordinary point nd a power series solution of a second order differential equation around a regular singular point using the method of Frobenius Overview for Exam 3 Sections 51 567l 759l 93l 0 52 Phase Plane 2 or more dimensions Critical points of autonomous system d T 101179 d d i 102179 are found by calculating the roots of f1Iy 0 f2zy In two dimensions the main types of behavior are 1 Node stable or unstable 2 Spiral stable or unstable 3 Saddle unstable In three dimensions there are many other types of behavior possible in cluding chaotic 0 53 Elimination Methods for constant coef cient equations Write the system in terms of D ddz and then eliminate all but one of the equations using algebraic methods M equations of order N will be reduced to a signel M X N equation usually 0 54 Coupled Spring Mass Systems System of equations comes from forcebalance lawsi Normal modes come from complex exponential solutions of resulting equa tions using elimination techniques of 53 Roots come in complex pairs 55 Electical Circuits Khirchoffls two laws conservation of chargecurrent and voltage drops give system of equations of the same form as massspring systemi 0 56 Numerical Methods for Higher Order Equations and Systems Write the equation as a system of rst order equations and then apply one of the following methods 1 Euler 2 Improved Euler 3i Runge Kutta For simple equations7 eigi y y y 1116 1 ND 1 40 2 Euler or Runge Kutta can be done by hand ie by calculatori 0 7 2 De nitiion of Laplace Transform Find the Laplace Transform of a continuous or pieceWise continuous function by means of its de nition um Me s mdt A table of Laplace transforms Will be provided on the exami 0 73 Properties of the Laplace Transform ll Linearity 2 Action on Derivatives 3i Multiplication by 6 quot 4i Multiplication by t 0 7 4 lnverse Laplace Transform H i Apply transform to ode i Solve for Ys Find inverse transform 9010 a Combine common terms Put fractions into standard form Use Method of Partial Fractions Noramlize leading coef cient in denominator Complete the square7 s 7 a2 2 Normalize numerator7 in terms of s 7 a Table lookup A AAAA m Gum CL 0 0quot VVVVVV 0 7 5 Solving lnitial Value Problems lncorporate initial conditions into Ysi 0 9 1 Matrix Methods Systems in Normal Form 0 92 Linear Algebraic Equations Gauss Jordan row reduction method Normal forms upper triangular diagonal 0 93 Matrices and Vectors Algebra of Matrices addition scalar multiplication matrixmatrix mul tiplicationi Noncommutativity of matrixvector multiplication Find determinanti Find inverse by rowreductioni

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