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# Introduction to Ordinary Differential Equations MATH 340

CSU

GPA 3.78

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This 4 page Class Notes was uploaded by Melvina Keeling on Monday September 21, 2015. The Class Notes belongs to MATH 340 at Colorado State University taught by Travis Olson in Fall. Since its upload, it has received 23 views. For similar materials see /class/210087/math-340-colorado-state-university in Mathematics (M) at Colorado State University.

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Date Created: 09/21/15

4234 Given a second order differential homogeneous equation x dm l sz 0 know the characteristic polynomial A2 dAw pA 7 27 2 and what the roots mean A13 W lf id274wg is real H12 7 444 344 0 then the homogeneous solution is simply z 15 1 025 If d274w3 0 then you have one repeating root A 7 and the homogeneous solution is z e dt2cl 0225 If H12 7 44 is imaginary call w x4w0 7 12 then the homogeneous solution is z e dt2clcoswt Cgsinwt If you have no damping d 0 then you get h 61COSw0t CgSiHltw0tgt 456 Given a general second order differential equation x dm wax ft lf ft Ptequot where Pt is a polynomial of order m the particular solution is mp pte where pt is a polynomial of order m whose coefficients are uniquely determined by solving mg 0le7 h1ng Pte m if ft Pt you just have T 0 If ft 5 Ptcoswt Qtsinwt let m be the maximum order of Pt and Qt then mp equotLptcoswt qtsinwt where pt and qt are both polynomials of order m that are uniquely determined by solving mg 0le7 h1ng 5 Ptcoswt Qtsinwt 47 Special case of above 51 Know de nition of Laplace transform fs Fs limTH00 fOT e gtftdt 52 Know the basic properties of the laplace transform and be able to apply them linearity reality y 3Ys7y0 yH 32Ys7sy07 y 0 edft Fs7 c tkft 71kFks Know how to use a table of laplace transforms and these properties to transform an ODE into an algebraic equation for Ys and solve for Ys 53 Know how to use a table of inverse transforms to turn Ys into lf Ys P where Ps and Qs are polynomials in s and A is a root of Qs with multiplicity m call QM 533 know Ys Bums with YA3 ff 931 where Aj di pjs Also if A is a complex root of Qs of multiplicity 1 A is also a root of multiplicity 1 then rams 153 2ReAe t 54 know how to put the previous section together to solve a second order initial value problem with a non zero forcing term 21 Know how to construct a model from a description of a physical situation Know how to nd the interval of existence of a solution to a rst order ODE Know what 3 means in terms of the physical situation and graph of solutions 3 tells you whether you are going positive or negative 22 Separable Equations Know how to solve an initial value problem for a separable differential equation 23 Models of Motion there were three main models here 1 no air resistance 2 linear air resistance 3 quadratic air resistance Know the solutions to all three and how to manipulate the solutions algebraically Also for more interesting systems cubic etc recall that these are separable autonomous differentiable equations and can be treated accordingly 24 Linear Equations Know how to use one of the techniques to solve a non homogeneous linear rst order ODE using an integrating factor or using variation of parameters 27 Existence and Uniqueness Know the theorems for the existence and uniqueness of solutions Know how to use the theorems to nd the maximum box R were you are guaranteed existence and uniqueness for a given initial value problem Know in particular that this means that solutions cannot cross in R and more speci cally that solutions cannot cross equilibrium solutions in R and how to use that to bound solutions between certain equilibrium values 29 Autonomous Equations and Stability Know how to nd equilibrium solutions in autonomous differential equations and how to determine the stability of those solutions Be able to draw approximate solutions of the behavior of various initial conditions ex see bottom of page 96 to see what I mean 31 Population Growth Know the meaning of the equilibrium points of the logistic problem and how to nd the new equilibrium points given harvesting For a given initial population and harvesting know how to gure out using the equilibrium points whether or not the population will survive over the long term 33 Financial models these are again linear differential equations 61 Euler s method know how to calculate a step or two of euler s method 71 Know how to do basic matrix and vector calculations Multiplication transpose addition and scaling 72 Know how to turn a linear system of equations into a matrix vector problem and solve simple ones 71 Know how to do matrix matrix and matrix vector and vector vector math mul tiplication addition subtraction transpose how to nd length of a vector multiplying by a scalar etc 72 know how to turn a system of linear equations into a matrix vector problem 73 Know how to augment a matrix Know how to nd solutions via row echelon or reduced row echelon form and how to recognize when a system has no solution last column in augmented matrix is a pivot column 74 Know the properties of null spaces of a matrix A and properties of subspaces in general 75 Know how to nd a basis for the null space of a given matrix Know that the basis elements of a null space need to be linearly independent and what it means to be linearly independent What a span of a set of vectors is spani 1i 2 fk is the set of all possible linear combinations of those vectors 76 A is a n x 71 square matrix Know when A is singular Af has a solution for any choice of I or equivalently the columns of A are linearly independent vectors or equivalently the rrefA In and why it is nice A 1 exists Know how to use augmentation and rref to nd A 1 mm1171 1mm 77 Know how to calculate the determinant of a 2 x 2 matrix and an upper or lower triangular matrix Know generic properties of the determinant detABdetAdetB detA 11detA A collection of n vectors E1 E2 mi is linearly independent if and only if XE1 E2 has detX7 0 ie X is non singular 81 Know how to turn a system of differential equations into a vector equation i Know how to turn a higher order equation into a system of rst order equations z m1 z 7 xi 2 and turn that system of equations into a vector equation 83 know how to nd nulclines and equilibrium points in systems of equations 84 Know how to recognize linear systems non linear systems homogeneous and non homogeneous systems Know how to turn linear systems both homogeneous and non homogeneous into vector matrix equations i At homogeneous if Be able to showdisprove that a suggested solution solves the given linear problem 85 Know that there are n fundamental solutions linearly independent for an n dimensional system and how to show that a suggested fundamental set is a fundamental set each vector solves the system and the wronskiandet 0 91 Know what an eigenvalueA and an eigenvector 17 is A17 A17 Know how to nd A and 17 for a 2x2 matrix A13 5 i xTZ 7 4D or by calculating detA7 AI0 and nding 17 as a basis element of nullA 7 AI 9234 Know how to classify the equilibrium point at the origin of 2 D linear systems as sources sinks saddles centers spirals nodes and if a spiral whether it goes clockwise counterclockwise in 2 V A by the trace T a d and determinant D ad 7 be of a matrix A Z Z based on D gt lt 0 T gt lt 0 and D lt gt T24 clockwise if c lt O counterclockwise c gt O 95 Know that real distinct eigenvalues lead to real linearly independent eigenvectors Complex eigenvalues come in pairs with the associated eigenvectors also coming in pairs A 7 17 177113 and A 7 17 177 If we have k real eigenvalues A1 Ak with k linearly independent eigenvec tors 171 17k and In complex eigenvectors Ak1 04k 1 k1Akm akm 1amp1quot with associated eigenvectors 17k1 17k1 1 13111 17k1m 17km 13km we also have the conjugates of these eigenvalues and eigenvectors for the system and fundamental solutions are E e Vt17j for 1 g j g k f eaft17jcos jt 7 w jsin jt and er eait17jsin jt w jcos jt 96 Know the de nition in class of the exponential of a matrix 5quot XtX0 1 where Xt is the fundamental matrix of solutions to AE i know the main properties of 5quot know what a generalized eigenvector is a vector 17 in the nullspace of A 7 AIk where k is the smallest integer such that dim null A 7 AIk is equal to the algebraic multiplicity of A for a 2 D system with one eigenvalue A and only one eigenvector 17 know that we nd a vector 13 such that A 7 AI1U 0117 and get fundamental solutions f1 5At17 and f2 e V1U 1 011725 97 Know that if a matrix has at least only negative eigenvectors then every solution tends towards the origin as t 7 00 if A has at least one positive eigenvalue then there exist trajectories that get arbitrarily far from the origin as t 7 00 Stable is if every solution that starts 77close77 to the origin stays 77close77 to the origin closed orbits are stable asymptotically stable means every solution decays to the origin as t 71nfty unstable is if it is not either one of the others 99 Know how to do variation of parameters for a non homogeneous system of equations re member your integration 4234 Given a second order or higher differential homogeneous equation know how to nd and solve for the characteristic polynomial and what the roots mean in terms of the homogeneous solution 45 Given a general second order linear differential equation y ay 1 by ft 1 if ft Pte where Pt is a polynomial of order m use the particular solution ypt pte where pt is a polynomial of order m whose coef cients are uniquely determined by solving 11 1 ay 1 byp Pte note if you just have ft Pt then you just have T 0 2 if ft 5quotPtcoswt Qtsmwt let m be the maximum order of Pt and Qt use the particular solution ypt e ptcoswt qtsmwt where both pt and qt are polynomials of order m Whose coefficients are uniquely determined by solving y Ly byp Pte 47 Given a second order differential equation y ay by ft where ft is a sinusoidal driving like coswt know how to calculate the gain in the particular solution due to the frequency of the driving and nd the frequency of the maximum gain 52 Know how to transform functions and differential equations via a Laplace transform yt a Ys and solve for Ys Be able to gure out what parts of the Ys equation come from the a initial conditions b driving and c homogeneous pieces 53 Be able to use algebra to massage the equations for Ys or Fs generically into terms you can inverse Laplace transform 5456 Be able to solve ODEs with possible discontinuous forcing by using Laplace transforms

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