{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 12 "Review Sheet" }}{PARA 257 "" 0 "" {TEXT -1 3 "and" }}{PARA 258 "" 0 "" {TEXT -1 16 "Practice Exam #2 " }}{PARA 259 "" 0 "" {TEXT -1 29 "Math 2250-3, November 6, 2000" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 290 12 "Review Sheet" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " Our exam covers chapters 4-5 of the text." }}{PARA 0 "" 0 "" {TEXT -1 205 " Only scientific calculators will be allowed on the \+ exam. But you can expect to be working with Maple output, in ways con sistent with the practice exam below and the homework problems you hav e worked." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 11 "Chapter 4: " }}{PARA 0 "" 0 "" {TEXT -1 123 " \+ At most 40% of the exam will deal directly with this material....but \+ much of Chapter 5 uses these concepts, so beware." }}{PARA 0 "" 0 "" {TEXT 261 17 "Know Definitions:" }}{PARA 0 "" 0 "" {TEXT -1 9 " (a ) " }{TEXT 257 13 "Vector Space:" }{TEXT -1 383 " A collection of obj ects which can be added and scalar multiplied, so that the usual arith emetic properties (Page 240) hold. You do not need to memorize all ei ght of these properties. The key point is that not only is R^n a vect or space, but also certain subsets of it are, and so are spaces made o ut of functions...because functions can be added and scalar multiplied (page 265.)" }}{PARA 0 "" 0 "" {TEXT -1 10 " (b) " }{TEXT 258 10 "Subspace: " }{TEXT -1 551 "a subset of a vector space which is its elf of vector space....to check whether a subset is actually a subspac e you only have to show that sums and scalar multiples of subset eleme nts are also in the subset (Theorem 1 page 242.) Examples of importan t subspaces are the set of homogeneous solutions to a matrix equation \+ (which I called the nullspace of the matrix and which the book calls t he solution space, page 243), the span of a collection of vectors (pag e 248), AND the set of homogenous solutions to a linear differential e quation (section 5.2)." }}{PARA 0 "" 0 "" {TEXT -1 11 " (c) A" } {TEXT 259 20 " linear combination " }{TEXT -1 93 "of a set of vectors \+ \{v1, v2, ...vn\} is any expression c1*v1 + c2*v2 + ... + cn*vn. (page 246)" }}{PARA 0 "" 0 "" {TEXT -1 14 " (d) The " }{TEXT 260 4 "sp an" }{TEXT -1 96 " of a set of vectors \{v1, v2, ...vn\} is the collec tion of all linear combinstations. (page 248)" }}{PARA 0 "" 0 "" {TEXT -1 42 " (e) A collection \{v1, v2, ...vn\} is " }{TEXT 262 18 "linearly dependent" }{TEXT -1 93 " if and only if some linear comb inatation (with not all ci's = 0) adds up to the zero vector." }} {PARA 0 "" 0 "" {TEXT -1 39 " (f) A collection \{v1, ... vn\} is \+ " }{TEXT 263 20 "linearly independent" }{TEXT -1 128 " if and only if \+ the only linear combination of them which adds up to zero is the one i n which all coefficients ci=0. (page 249)" }}{PARA 0 "" 0 "" {TEXT -1 12 " (g) A " }{TEXT 264 5 "basis" }{TEXT -1 137 " for a vector space (or subspace) is a set of vectors \{v1, ..., vk\} which space t he space and which are linearly independent. (page 255.)" }}{PARA 0 " " 0 "" {TEXT -1 14 " (h) The " }{TEXT 265 9 "dimension" }{TEXT -1 58 " of a vector space is the number of elements in any basis." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 11 "Know Fac ts:" }}{PARA 0 "" 0 "" {TEXT -1 163 " (a) If the dimension of a v ector space is n, then no collection of fewer than n vectors can span \+ and every collection with more than n elements is dependent." }}{PARA 0 "" 0 "" {TEXT -1 180 " (b) n vectors in R^n are a basis if and \+ only if the square matrix in which they are the columns is non-singula r. So you can use det or rref as a test for basis in this case." }} {PARA 0 "" 0 "" {TEXT -1 111 " (c) Basically all linear independe nce and span questions in R^n can be answered using rref. (see below. )" }}{PARA 0 "" 0 "" {TEXT -1 156 " (d) You can toss dependent ve ctors out of a collection without changing the span. In this manner yo u can take a spanning set and turn it into a basis." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 16 "Do computations:" }} {PARA 0 "" 0 "" {TEXT -1 146 " (a) Be able to check whether vector s are independent or independent, e.g. problems page 254. Know how to use rref to check for dependencies." }}{PARA 0 "" 0 "" {TEXT -1 103 " (b) Be able to find bases for the solution space to homogeneous \+ equations, e.g. problems page 262" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 268 12 "Chapter 5: " }}{PARA 0 "" 0 "" {TEXT -1 201 " At least 60% of the exam will cover this material, and at least 40% of it will be from sections 5.4 and 5.5. (Answering questi ons from 5.4 and 5.5 almost always uses 5.1-5.3 material implicitly.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 29 "5.1- 5.3, 5.5 General theory:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 269 29 " Linear differential equation" }{TEXT -1 12 " (page 287.)" }} {PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 270 26 "principle of s uperposition" }{TEXT -1 371 " (e.g. Theorem 1 page 287, also leads to \+ the fact that the general solution y to the inhomogeneous equation is \+ yp + yh, where yp is a particular solution, and yh is the general solu tion to the homogeneous equation. (Theorem 5 page 296.) Also leads t o a method for getting particular solutions which are sums of particu lar solutions for pieces of the right hand side.) " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 272 11 "homogeneous" }{TEXT -1 148 " \+ (L(y)=0). Solution space is an n-dimensional vector space. Know how to find it for constant coefficients, using exponentials and the resu lting " }{TEXT 276 23 "characteristic equation" }{TEXT -1 6 " and " } {TEXT 275 13 "Euler formula" }{TEXT -1 202 " if necessary (section 5. 3 and problems). What to do with repeated roots. The Wronskian test \+ for linear independence will not be on the exam, although you should k now it, and understand why it works." }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }{TEXT 273 13 "inhomogeneous" }{TEXT -1 173 " (L(y)=f). \+ Know how to find particular solutions by the method of undetermined co efficients. Variation of parameters will not be on the exam. (Sectio n 5.5 and problems)" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{TEXT 274 51 " initial value problem, existence and uniqueness. " }{TEXT -1 135 "Know how to solve initial value problems by finding yp, and yh , and then finding values of constants in yh to match initial conditio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 59 "5 .4 and 5.6: Mechanical vibrations and forced oscillations:" }}{PARA 0 "" 0 "" {TEXT -1 38 " xh (homogeneous solutions) cases:" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 279 8 "undamped" }{TEXT -1 131 " (simple harmonic motion)\n going from A*cos(wt) + \+ B*cos(wt) to C*cos(wt-a). (The ABC triangle, amplitude and phase.)" } }{PARA 0 "" 0 "" {TEXT -1 78 " derivation of spring equa tion from Newton's and Hooke's Laws. " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 278 7 "damped." }}{PARA 0 "" 0 "" {TEXT -1 14 " \+ " }{TEXT 280 45 " under-damped, over-damped, critically dam ped" }{TEXT -1 62 ". Know how to recognize, and different forms of th e solution." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 281 20 "force d oscillations:" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 282 9 "undamped:" }}{PARA 0 "" 0 "" {TEXT -1 15 " " } {TEXT 283 9 "resonance" }{TEXT -1 78 ", and when it arises. form of s olution, as follows from general theory above." }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{TEXT 284 7 "beating" }{TEXT -1 46 " wi ll not be on the exam, but it is important." }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 285 7 "damped:" }}{PARA 0 "" 0 "" {TEXT -1 42 " general solution is sum of " }{TEXT 286 21 "steady \+ state periodic" }{TEXT -1 7 ", with " }{TEXT 287 9 "transient" }{TEXT -1 45 ". How to find each piece, and express in the" }}{PARA 0 "" 0 " " {TEXT -1 22 "amplitude- phase form." }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{TEXT 288 22 "practical resonance. " }{TEXT -1 79 " will occur if damping is small and driving frequency is near natural f requency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 289 13 "Practice Exam" }} {PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "1) Consider the homogeneou s differential equation" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&d eqtnG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\")F2-F(6$F*F- F2F2*&\"#?F2F*F2F2\"\"!" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 320 "1a) If this was modeling a mass-spring configura tion like we studied in Chapter 5 of Edwards-Penney, and if the mass w as 3 kg, what values of coefficient of friction and spring constant wo uld lead to the differential equation above? (1 point for getting the units correct, 2 points for the correct numerical values). " }} {PARA 262 "" 0 "" {TEXT -1 10 "(6 points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "1b) What kind of damping is exhib ited by this mass-spring system?" }}{PARA 263 "" 0 "" {TEXT -1 10 "(4 \+ points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "1c) Find the general solution to this homogeneous differential eq uation" }}{PARA 265 "" 0 "" {TEXT -1 10 "(5 points)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 334 "1d) Consider the same spring system, but now with a driving force F0(t)=9*cos(2t). Find t he general solution to this inhomogeneous differential equation. Use \+ the method of undetermined coefficients. Identify the steady periodic and transient pieces of the solution. Find the amplitude and phase \+ of the steady periodic solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 264 "" 0 "" {TEXT -1 11 "(20 points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "2) Here is a matrix:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'\"\"\"\"\"$!\"%! \")\"\"'7'F*\"\"!\"\"#F*F+7'F1\"\"(!#5!#>\"#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Here is its reduced row echelon form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7'\"\"\"\"\"!\"\"#F(\"\"$7'F)F(!\"#!\"$F( 7'F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 76 "2a) Find a basis for th e solution space (of homogeneous solutions) to Ax=0." }}{PARA 266 "" 0 "" {TEXT -1 11 "(10 points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "2b) Explain what it means for a collect ion of vectors to be linearly dependent or linearly independent." }} {PARA 268 "" 0 "" {TEXT -1 10 "(5 points)" }}{PARA 0 "" 0 "" {TEXT -1 166 "2c) Are the first three columns of A linearly independent or lin early dependent? If they are dependent, exhibit a dependency. If the y are independent, explain why." }}{PARA 267 "" 0 "" {TEXT -1 11 "(10 \+ points)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "2d) Explain what it means for a collection of vectors to span a v ector space." }}{PARA 269 "" 0 "" {TEXT -1 10 "(5 points)" }}{PARA 271 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "2e) Do the f irst three columns of A span all of R^3. Explain your answer." }} {PARA 272 "" 0 "" {TEXT -1 11 "(10 points)" }}{PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "3) Consider the differential equation" }}{PARA 270 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqtnG/,&-%%diffG6$- %\"yG6#%\"xG-%\"$G6$F-\"\"$\"\"\"*&\"#DF2-F(6$F*F-F2F2\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 117 "Find the solution to the initial value problem for this differential equation, with y(0)=4, D(y)(0)=0, D(D(y)(0))=10 ." }}{PARA 274 "" 0 "" {TEXT -1 11 "(25 points)" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}}{MARK "102 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }