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# Applied Differential Equations I MA 341

NCS

GPA 3.93

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This 8 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 341 at North Carolina State University taught by Stephen Schecter in Fall. Since its upload, it has received 30 views. For similar materials see /class/223712/ma-341-north-carolina-state-university in Mathematics (M) at North Carolina State University.

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Date Created: 10/15/15

MA 341 003 Test 1 Review Questions S Soheoter September 14 2006 Warning Not everything is covered H F 9 7 U 03 5 00 p Translate into a differential equation Sec 11 problems 15 and 16 Checking whether something is a solution of a differential equation Sec 12 problems 8 and 10 Existence uniqueness theorem Sec 12 problem 26 Also What if the initial condition is y1 2 Direction elds Sec 13 problem 17 Use the method of isoclines to sketch the direction eld in the region z gt 0 Then try to sketch the solutions with y1 1 and y1 5 For these two solutions7 as x a 007 what do you think y approaches Euler7s method Sec 14 problem 6 Just do the points z 117 12 Don7t round Separable equations Sec 22 problem 26 Linear equations Sec 23 problems 167 20 Exact equations Sec 24 problem 12 Mixing Sec 32 problem 2 Mechanics Sec 34 problem 6 Initially the object is 100 m above the ground First draw your coordinate axis Answers 1 3 9 7 U 5 Both are solutions Just calculate 16 kAZ dy dm and plug into the differential equation No7 yes Look at at the two points Suggestion Look at the isoclines c 72 71 07 17 2 Approximation from Euler7s method At z 117y 01 at z 127y 0209 y 17ln12 16 y lt z270gt 20 yz27 emsiny7x3y C z 25 7 205122 t 1M 5776 If x increases as you go up7 1 749 2496 If the initial position of the object is z 07 z 749t 7 1245641 1245 Then z 7100 when after simplifying 2295 t 2546 Solution according to Maple t 2295 Not surprising H D 00 r U 03 1 MA 341 005 Test 2 Review Questions S Soheoter March 16 2005 Undetermined coef cients Sec 44 problems 18 and 327 Sec 45 problems 18 and 28 Variation of parameters Sec 46 problem 18 Mass spring systems Sec 48 problem 2 Laplace transforms Sec 73 problems 6 and 10 lnverse Laplace transforms Sec 74 problems 24 and 26 Solving initial value problems Sec 75 problem 11 Transforms of discontinuous and periodic functions Sec 76 problems 28 y sint for 0lttlt7r32 Answers 1 F 9 7 U 5 44 problem 18 yp 72tcos 2t 44 problem 32 tAt6 Bt5 Ot4 Dt3 Etz Ft Ge 3t 45 problem 18 0163 026 7 t2 47 45 problem28 f727756 721 46 problem 18 0163 02125 g 48 problem 2 y 7icos 5t 7 sin 5t sin 5t b where b arctan 7 7T 72246 amplitude period 2quot frequency passes through equilibrium at t w 0449 seconds 73 problem 6 W 9722725 73 problem 10 W 74 problem 24 56 262 cos 3t 7 562 sin 3t 74 problem 26 1 7 gt2 7 662 75 problem 11 see text 76 problem 28 W 76 problem 32 cost 7 sin 2t 7 2 sin tut 7 27139 7 sin 2tut 7 27139 MA 341 001 Sorne Review Questions on Secs 91796 S Scheoter Updated April 12 2006 1 Consider the differential equation 7 x t 7 i4 0 xt X1t cos2tt and X2tsin2t39 72sin2 200s 2t a Check that x1t is a solution Just substitute and Check that it works b x2t is also a solution You dont need to Check this Show that x1t and x2t are linearly independent 0 Give the general solution d Give the solution that satis es the initial condition f Give the general solution 2 Find the eigenvalues of the matrix A DHM NW0 HWO 3 The matrix 0 1 2 A 1 2 0 0 71 0 has the eigenvalues r 71 17 2 An eigenvector for the eigenvalue 71 is 73 1 1 a Find an eigenvector for the eigenvalue 1 b Find an eigenvector for the eigenvalue 2 c Give the general solution of x t Axt A 134 ll has the eigenvalues r 1 i 2239 Find the general solution of x t Axt 4 The matrix Sorne answers cos 2t sin 2t 139 C Kat Cl 2 sin 2t CZ 2 cos 2t 3 cos 2t 7 sin 2t 2 d X 6 sin 2t 7 3 cos 2t cos 2t sin 2t 6 f X 7 Cl 2 sin 2t CZ 2 cos 2t 7464 2 2 213239 1 0 3 a 71 b 72 1 1 C 73 1 0 Xt cle t 1 026 1 6362 72 1 1 1 X0376 etsin2t 6 fetcos2t 7 1 2etcos2t 2 2etsin2t H E0 00 r U 03 I 00 K O MA 341 003 Test 2 Review Questions S Schecter October 26 2006 Homogeneous 2nd order linear equations with constant coef cients Sec 42 problems 47 16 sec 437 problems 147 24 Undetermined coef cients Sec 44 problems 18 and 327 Sec 45 problems 18 and 28 Variation of parameters Sec 46 problem 18 Mass spring systems Sec 48 problem 2 De nition of Laplace transform Sec 72 problem 4 Laplace transforms from tables Sec 73 problems 6 and 10 lnverse Laplace transforms Sec 74 problems 24 and 26 Solving initial value problems Sec 75 problem 10 Transforms of discontinuous functions Sec 76 problem 32 Transforms of delta functions Use last homework assignment Answers 1 F 9 7 U 03 5 00 H 0 H H H D H 9 H r H CH H 03 72 problem 4 1 74 problem 26 42 problem 4 016 Cgte gt 42 problem 16 A6 7 163 43 problem 14 016 cos 5t 026 sin St 43 problem 24 i sin 3t cos 3t 44 problem 18 yp 72tcos 2t 44 problem 32 tAt6 Bt5 Ct4 D151 Etz Ft Ge 3t 45 problem 18 0163 026 7 t2 47 45problem28 777 46 problem 18 0163 cztei g 48 problem 2 y 7 cos 5t 7 sin 5t sin 5t b where b arctan 7 7T 72246 E 0 7 0449 seconds amplitude period 2quot frequency 7 passes through equilibrium at t 7r7arctan 5 sgt3 57327 73 problem 6 2 52393 9224 972 2725 73 problem 10 W 74 problem 24 56 262 cos 3t 7 562 sin 3t 1 i gt 652 75 problem 10 7t 7 6 2 2te 2t 62 76 problem 32 cost 7 sin 2t 7 2 sin tut 7 27139 sin 2tut 7 27139

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