{"id":"5c8199d3b8497d000d4e5eb1","name":"Notes on Diffy Qs (Lebl)","summary":"Edfinity is supported by the National Science Foundation","description":"\u003cp dir=\"ltr\"\u003eThis is an online homework companion to \u003ca href=\"https://www.jirka.org/diffyqs/\"\u003e\u003cem\u003eNotes on Diffy Qs: Differential Equations for Engineers\u003c/em\u003e\u003c/a\u003e\u003cem\u003e\u0026nbsp;by\u003c/em\u003e Jiri Lebl. It comprises hundreds of algorithmic problems carefully organized into problem sets mapped to textbook sections. Use this course as-is, or customize at any level. You can mix-and-match problems from other catalog courses, add problems from the Edfinity problem repository, or write your own\u003c/p\u003e\u003cp dir=\"ltr\"\u003e\u003cstrong\u003eHow to use this course\u003c/strong\u003e\u003c/p\u003e\u003col\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eHomework\u003c/strong\u003e\u003cstrong\u003e:\u003c/strong\u003e Assign high quality problems with hints and personalized feedback to develop problem-solving skills.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eTesting\u003c/strong\u003e\u003cstrong\u003e:\u003c/strong\u003e Create summative secure online quizzes and tests in minutes.\u0026nbsp;\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eSupplementary resources:\u003c/strong\u003e Embed videos, class notes, and applets alongside assignments.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eIntervention:\u003c/strong\u003e Use rich analytics to identify and monitor at-risk students for timely intervention.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eAnalytics:\u003c/strong\u003e Drill down into student performance and identify problematic or difficult topics.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eFlipped classroom:\u003c/strong\u003e Assign pre-class assignments. Save precious class time for discussions.\u003c/li\u003e\u003cli dir=\"ltr\"\u003e\u003cstrong\u003eEmporium classes:\u003c/strong\u003e Use Edfinity for individual/group work for large enrollment sections in labs.\u003c/li\u003e\u003c/ol\u003e","description_more":"","price":29.0,"sort_order":0,"active":true,"individual_license":false,"section_id":"5dd05e13c5b59e00046bd771","permissive_license":false,"type":"course","year":null,"section_license":true,"taxable":false,"purchasable":true,"related_product_ids":[],"paired_product_id":null,"min_quantity":null,"max_quantity":null,"audience":"educator_audience","site_ids":["55843bd46435650003000000"],"package_price":null,"show_toc":true,"author_roles":[],"author_role_title":"Contributors","institution":null,"tags_array":["math","oer","differential_equations"],"hidden_tags":[],"meta_description":null,"term_pricing":true,"curricula":null,"curricula_course":null,"highlights":"\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-subscript fa-fw mr-2\"\u003e\u003c/i\u003e \u003cstrong\u003eInteractive, algorithmic problems\u003cbr\u003e\u003c/strong\u003eAlgebraic, graphing, open response; randomized variants, hints, and tips\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-hand-holding-heart fa-fw mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eAdaptive learning and personalization\u003cbr\u003e\u003c/strong\u003eEach student receives personalized support\u003cbr\u003eto fill learning gaps.\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-fw fa-code-branch mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eCorequisite course structures\u003cbr\u003e\u003c/strong\u003eUse pre-built corequisite content,\u003cbr\u003eor create your own.\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-hands-helping fa-fw mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eLMS integration and accessibility\u003cbr\u003e\u003c/strong\u003eConnect to your LMS in minutes. \u003ca class=\"”underline”\" href=\"http://edfinity.com/vpat\" target=\"_blank\"\u003eVPAT here\u003c/a\u003e.\u003c/div\u003e\n\u003cdiv class=\"mb-3\"\u003e\u003ci class=\"fa fa-plug fa-fw mr-2\"\u003e\u003c/i\u003e\u003cstrong\u003eWeBWorK-compatible\u003cbr\u003e\u003c/strong\u003eImport and author WeBWorK problems.\u003c/div\u003e\n","preview_assignment":{"id":"5dd05e13c5b59e00046bd779","name":"Sec 0.2:  Introduction to differential equations","product":{"id":"5c8199d3b8497d000d4e5eb1","name":"Notes on Diffy Qs (Lebl)"},"master_set":{"id":"61d51c51abeba400048cac9f","problems":{"id":"611a591875b85800033de89e","proof":false,"graphical_geometric":false,"tags_array":[],"grade_level_from":null,"grade_level_to":null,"skill_ids":["5a141faeac370c0004c77434"],"contest_event_ids":[],"header":"","solution":"","part_attributes":[{"id":"5d10e4703b29c40004ba3ded","part_id":"5d10e4703b29c40004ba3dec","description":null,"points":7.0,"automatic_manual_credit":true,"format":"ww","pg":"##DESCRIPTION\n## Plugging a test solution into a differential equation, verifying it, and verifying a nonsolution\n##ENDDESCRIPTION\n\n## DBsubject(Differential equations)\n## DBchapter(Introductory concepts)\n## DBsection(Verification of solutions)\n## Institution(Fort Lewis College)\n## Author(Paul Pearson)\n## Level(3)\n## TitleText1('Notes on Diffy Qs: Differential Equations for Engineers')\n## AuthorText1('Jiri Lebl')\n## EditionText1('6')\n## Section1('0.2')\n## Problem1('')\n## MO(1)\n## KEYWORDS('ODE', 'ordinary', 'differential', 'equation', 'verify')\n\n##############################\n#  Initialization\n\nDOCUMENT(); \n\nloadMacros(\n  \"PGstandard.pl\",\n  \"MathObjects.pl\",\n  \"AnswerFormatHelp.pl\",\n  \"parserPopUp.pl\",\n  \"answerHints.pl\",\n  \"PGcourse.pl\"\n);\n\nTEXT(beginproblem());\n\n#############################\n#  Setup\n\nContext(\"Numeric\")-\u003evariables-\u003eadd(\n  t=\u003e\"Real\"\n);\n\n$k = random(2,5,1);\n$k3 = 3 * $k;\n$kk3 = 3 * $k**2;\n$kkk = $k**3;\n$x = Formula(\"e^($k t)\");\n$expr = \"x''' - $k3 x'' + $kk3 x' - $kkk x\";\n\nContext()-\u003eflags-\u003eset(\n  reduceConstants=\u003e0, # no decimals\n  reduceConstantFunctions=\u003e1, # combine 4+5*2?\n  formatStudentAnswer=\u003e'parsed', # no decimals\n);\n\n$answer[0] = Formula(\"$k^3 * $x\")-\u003ereduce;\n$answer[1] = Formula(\"-3*$k * $k^2*$x\")-\u003ereduce;\n$answer[2] = Formula(\"3*$k^2 * $k*$x\")-\u003ereduce;\n$answer[3] = Formula(\"-$k^3*$x\")-\u003ereduce;\n\n$answer[4] = Formula(\"0\");\n\n$answer[5] = PopUp([\"Choose\",\"Yes\",\"No\"],\"Yes\");\n\n$answer[6] = PopUp([\"Choose\",\"Yes\",\"No\"],\"No\");\n\n#############################\n#  Main text\n\nContext()-\u003etexStrings;\nBEGIN_TEXT\nSuppose \\( x = $x \\).  \n\n$PAR\nFind the value of the expression\n\\( $expr \\) in terms of the variable \\( t \\).\n$BR\n$BITALIC Enter the terms in the order given. $EITALIC\n$PAR\n\\{ ans_rule(10) \\}\n+\n\\{ ans_rule(10) \\}\n+\n\\{ ans_rule(10) \\}\n+\n\\{ ans_rule(10) \\}\n\\{ AnswerFormatHelp(\"formulas\") \\}\n\n$PAR\nSimplify your answer above to obtain\na differential equation in terms\nof the dependent variable \\( x \\) satisfied\nby \\( x = $x \\).  \n$PAR\n\\( $expr = \\) \n\\{ ans_rule(10) \\}\n\\{ AnswerFormatHelp(\"formulas\") \\}\n\n$PAR\nSo, is \\(x = $x \\) a solution to your\ndifferential equation above?\n\\{ $answer[5]-\u003emenu() \\}\n\n$PAR\nNow,\nis \\( x = e^t \\) a solution to your\ndifferential equation above?\n\\{ $answer[6]-\u003emenu() \\}\n$BR\n$BITALIC Be sure you can\njustify your answer. $EITALIC\nEND_TEXT\nContext()-\u003enormalStrings;\n\n############################\n#  Answers\n\n$showPartialCorrectAnswers = 1;\n\nANS( $answer[0]-\u003ecmp() );\nANS( $answer[1]-\u003ecmp()-\u003ewithPostFilter(AnswerHints(\n  [Compute(\"-$answer[1]\"),\n   Compute(\"-1/(3*$k)*$answer[1]\"),\n   Compute(\" 1/(3*$k)*$answer[1]\")] =\u003e \n  \"Check your signs and the coefficients in the D.E.\",\n)) \n);\nANS( $answer[2]-\u003ecmp()-\u003ewithPostFilter(AnswerHints(\n  [Compute(\"1/(3*$k^2)*$answer[2]\")] =\u003e \n  \"Check your signs and the coefficients in the D.E.\",\n)) \n);\nANS( $answer[3]-\u003ecmp()-\u003ewithPostFilter(AnswerHints(\n  [Compute(\"-$answer[3]\"),\n   Compute(\"-1/($k^3)*$answer[3]\"),\n   Compute(\" 1/($k^3)*$answer[3]\")] =\u003e \n  \"Check your signs and the coefficients in the D.E.\",\n))  \n);\nANS( $answer[4]-\u003ecmp() );\nANS( $answer[5]-\u003ecmp() );\nANS( $answer[6]-\u003ecmp() );\n\n############################\n#  Solution\n\nContext()-\u003etexStrings;\nBEGIN_SOLUTION\nAs \\(x = $x ,\\) then\n\\( x' = $k $x ,\\)\n\\(x'' = $k^2 $x ,\\) and\n\\(x''' = $k^3 $x . \\)\nSo the terms are:\n\\[x''' = $answer[0] , \\]\n\\[- $k3 x'' = $answer[1] , \\]\n\\[$kk3 x' = $answer[2] , \\]\n\\[-$kkk x = $answer[3] . \\]\nWhen we add those terms together we get \\(0\\), and hence the differential equation we get is\n\\[x''' - $k3 x'' + $kk3 x' - $kkk x = 0 .\\]\nThe right hand side \\(0\\) is precisely what we get when we plug in \\(x=$x,\\)\nand so \\(x = $x\\) is a solution to the equation.\n$PAR\nOn the other hand when we plug in \\(x = e^x\\) to the left hand side we get\n\\[ x''' - $k3 x'' + $kk3 x' - $kkk x = e^x - $k3 e^x + $kk3 e^x - $kkk e^x = (1-$k3+$kk3-$kkk) e^x \\not= 0 .\\]\nSince the left hand side is not zero when we plug in \\(x = e^x,\\)\nthen this is not a solution to the differential equation.\nEND_SOLUTION\nContext()-\u003enormalStrings;\n\n############################\n\nCOMMENT(\"Plugging a test solution into a differential equation, verifying it, and verifying a nonsolution.\");\n\nENDDOCUMENT();","use_equation_editor":null,"webwork_features":[],"part_attributes":[{"id":"5dd05e14c5b59e00046bd780","part_id":"5d10e4703b29c40004ba3def","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"125*e^(5*t)","format":"ws","answer_name":"AnSwEr0001","extra":false,"applet_state":false,"multipart":true,"manually_gradable":false,"answer_text":"\\(125e^{5t}\\)"},{"id":"5dd05e14c5b59e00046bd781","part_id":"5d10e4703b29c40004ba3df0","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"(-375)*e^(5*t)","format":"ws","answer_name":"AnSwEr0002","extra":false,"applet_state":false,"multipart":true,"manually_gradable":false,"answer_text":"\\((-375)e^{5t}\\)"},{"id":"5dd05e14c5b59e00046bd782","part_id":"5d10e4703b29c40004ba3df1","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"375*e^(5*t)","format":"ws","answer_name":"AnSwEr0003","extra":false,"applet_state":false,"multipart":true,"manually_gradable":false,"answer_text":"\\(375e^{5t}\\)"},{"id":"5dd05e14c5b59e00046bd783","part_id":"5d10e4703b29c40004ba3df2","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"-125*e^(5*t)","format":"ws","answer_name":"AnSwEr0004","extra":false,"applet_state":false,"multipart":true,"manually_gradable":false,"answer_text":"\\(-125e^{5t}\\)"},{"id":"5dd05e14c5b59e00046bd784","part_id":"5d10e4703b29c40004ba3df3","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"0","format":"ws","answer_name":"AnSwEr0005","extra":false,"applet_state":false,"multipart":true,"manually_gradable":false,"answer_text":"\\(0\\)"},{"id":"5dd05e14c5b59e00046bd785","part_id":"5d10e4703b29c40004ba3df4","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"Yes","format":"ws","answer_name":"AnSwEr0006","extra":false,"applet_state":false,"multipart":true,"manually_gradable":false,"answer_text":"Yes"},{"id":"5dd05e14c5b59e00046bd786","part_id":"5dd05e14c5b59e00046bd77f","description":null,"points":1.0,"automatic_manual_credit":true,"answer":"No","format":"ws","answer_name":"AnSwEr0007","extra":false,"applet_state":false,"multipart":false,"manually_gradable":false,"answer_text":"No"}],"interactive_html":"\nSuppose \\(x = e^{5t}\\).  \n\n\u003cp\u003e\nFind the value of the expression\n\\(x''' - 15 x'' + 75 x' - 125 x\\) in terms of the variable \\(t\\).\n\u003cbr\u003e\n\u003ci\u003e Enter the terms in the order given. \u003c/i\u003e\n\u003c/p\u003e\n\u003cp\u003e\n\u003cinput type=\"text\" class=\"codeshard\" size=\"10\" name=\"AnSwEr0001\" id=\"AnSwEr0001\" aria-label=\"answer 1 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0001\" value=\"\"\u003e\n\n+\n\u003cinput type=\"text\" class=\"codeshard\" size=\"10\" name=\"AnSwEr0002\" id=\"AnSwEr0002\" aria-label=\"answer 2 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0002\" value=\"\"\u003e\n\n+\n\u003cinput type=\"text\" class=\"codeshard\" size=\"10\" name=\"AnSwEr0003\" id=\"AnSwEr0003\" aria-label=\"answer 3 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0003\" value=\"\"\u003e\n\n+\n\u003cinput type=\"text\" class=\"codeshard\" size=\"10\" name=\"AnSwEr0004\" id=\"AnSwEr0004\" aria-label=\"answer 4 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0004\" value=\"\"\u003e\n\n\u003ca href=\"#\" onclick=\"return openhelpFormulas()\"\u003ehelp (formulas)\u003c/a\u003e\n\n\u003c/p\u003e\n\u003cp\u003e\nSimplify your answer above to obtain\na differential equation in terms\nof the dependent variable \\(x\\) satisfied\nby \\(x = e^{5t}\\).  \n\u003c/p\u003e\n\u003cp\u003e\n\\(x''' - 15 x'' + 75 x' - 125 x =\\) \n\u003cinput type=\"text\" class=\"codeshard\" size=\"10\" name=\"AnSwEr0005\" id=\"AnSwEr0005\" aria-label=\"answer 5 \" dir=\"auto\" value=\"\"\u003e\n\u003cinput type=\"hidden\" name=\"previous_AnSwEr0005\" value=\"\"\u003e\n\n\u003ca href=\"#\" onclick=\"return openhelpFormulas()\"\u003ehelp (formulas)\u003c/a\u003e\n\n\u003c/p\u003e\n\u003cp\u003e\nSo, is \\(x = e^{5t}\\) a solution to your\ndifferential equation above?\n\u003cselect name=\"AnSwEr0006\" id=\"AnSwEr0006\" aria-label=\"answer 6 \" size=\"1\" class=\"base-select\"\u003e\n\u003coption value=\"Choose\" class=\"tex2jax_ignore\"\u003eChoose\u003c/option\u003e\n\u003coption value=\"Yes\" class=\"tex2jax_ignore\"\u003eYes\u003c/option\u003e\n\u003coption value=\"No\" class=\"tex2jax_ignore\"\u003eNo\u003c/option\u003e\n\u003c/select\u003e\n\n\u003c/p\u003e\n\u003cp\u003e\nNow,\nis \\(x = e^t\\) a solution to your\ndifferential equation above?\n\u003cselect name=\"AnSwEr0007\" id=\"AnSwEr0007\" aria-label=\"answer 7 \" size=\"1\" class=\"base-select\"\u003e\n\u003coption value=\"Choose\" class=\"tex2jax_ignore\"\u003eChoose\u003c/option\u003e\n\u003coption value=\"Yes\" class=\"tex2jax_ignore\"\u003eYes\u003c/option\u003e\n\u003coption value=\"No\" class=\"tex2jax_ignore\"\u003eNo\u003c/option\u003e\n\u003c/select\u003e\n\u003cbr\u003e\n\u003ci\u003e Be sure you can\njustify your answer. \u003c/i\u003e\n\n\u003c/p\u003e\n\u003cdiv class=\"solution\"\u003e\n\u003cp\u003e\u003cb\u003eSolution:\u003c/b\u003e \u003cbr\u003eAs \\(x = e^{5t} ,\\) then\n\\(x' = 5 e^{5t} ,\\)\n\\(x'' = 5^2 e^{5t} ,\\) and\n\\(x''' = 5^3 e^{5t} .\\)\nSo the terms are:\n\\[x''' = 125e^{5t} ,\\]\n\\[- 15 x'' = (-375)e^{5t} ,\\]\n\\[75 x' = 375e^{5t} ,\\]\n\\[-125 x = -125e^{5t} .\\]\nWhen we add those terms together we get \\(0\\), and hence the differential equation we get is\n\\[x''' - 15 x'' + 75 x' - 125 x = 0 .\\]\nThe right hand side \\(0\\) is precisely what we get when we plug in \\(x=e^{5t},\\)\nand so \\(x = e^{5t}\\) is a solution to the equation.\n\u003c/p\u003e\n\u003cp\u003e\nOn the other hand when we plug in \\(x = e^x\\) to the left hand side we get\n\\[x''' - 15 x'' + 75 x' - 125 x = e^x - 15 e^x + 75 e^x - 125 e^x = (1-15+75-125) e^x \\not= 0 .\\]\nSince the left hand side is not zero when we plug in \\(x = e^x,\\)\nthen this is not a solution to the differential equation.\n\n\u003c/p\u003e\n\u003c/div\u003e","static_html":"\u003cp\u003e\nSuppose \\(x = e^{5t}\\).  \n\n\u003c/p\u003e\n\u003cp\u003e\nFind the value of the expression\n\\(x''' - 15 x'' + 75 x' - 125 x\\) in terms of the variable \\(t\\).\n\u003cbr\u003e\n\u003ci\u003e Enter the terms in the order given. \u003c/i\u003e\n\u003c/p\u003e\n\u003cp\u003e\n\u003cspan class=\"ww-blank\" name=\"AnSwEr0001\"\u003e \u003c/span\u003e\n\n\n+\n\u003cspan class=\"ww-blank\" name=\"AnSwEr0002\"\u003e \u003c/span\u003e\n\n\n+\n\u003cspan class=\"ww-blank\" name=\"AnSwEr0003\"\u003e \u003c/span\u003e\n\n\n+\n\u003cspan class=\"ww-blank\" name=\"AnSwEr0004\"\u003e \u003c/span\u003e\n\n\n\n\n\u003c/p\u003e\n\u003cp\u003e\nSimplify your answer above to obtain\na differential equation in terms\nof the dependent variable \\(x\\) satisfied\nby \\(x = e^{5t}\\).  \n\u003c/p\u003e\n\u003cp\u003e\n\\(x''' - 15 x'' + 75 x' - 125 x =\\) \n\u003cspan class=\"ww-blank\" name=\"AnSwEr0005\"\u003e \u003c/span\u003e\n\n\n\n\n\u003c/p\u003e\n\u003cp\u003e\nSo, is \\(x = e^{5t}\\) a solution to your\ndifferential equation above?\n\u003cselect class=\"pg-select\" name=\"AnSwEr0006\" id=\"AnSwEr0006\" aria-label=\"answer 6 \" size=\"1\"\u003e\n\u003coption value=\"Choose\" class=\"tex2jax_ignore\"\u003eChoose\u003c/option\u003e\n\u003coption value=\"Yes\" class=\"tex2jax_ignore\"\u003eYes\u003c/option\u003e\n\u003coption value=\"No\" class=\"tex2jax_ignore\"\u003eNo\u003c/option\u003e\n\u003c/select\u003e\n\n\u003c/p\u003e\n\u003cp\u003e\nNow,\nis \\(x = e^t\\) a solution to your\ndifferential equation above?\n\u003cselect class=\"pg-select\" name=\"AnSwEr0007\" id=\"AnSwEr0007\" aria-label=\"answer 7 \" size=\"1\"\u003e\n\u003coption value=\"Choose\" class=\"tex2jax_ignore\"\u003eChoose\u003c/option\u003e\n\u003coption value=\"Yes\" class=\"tex2jax_ignore\"\u003eYes\u003c/option\u003e\n\u003coption value=\"No\" class=\"tex2jax_ignore\"\u003eNo\u003c/option\u003e\n\u003c/select\u003e\n\u003cbr\u003e\n\u003ci\u003e Be sure you can\njustify your answer. \u003c/i\u003e\n\n\u003c/p\u003e\n\u003cdiv class=\"solution\"\u003e\n\u003cp\u003e\u003cb\u003eSolution:\u003c/b\u003e \u003cbr\u003eAs \\(x = e^{5t} ,\\) then\n\\(x' = 5 e^{5t} ,\\)\n\\(x'' = 5^2 e^{5t} ,\\) and\n\\(x''' = 5^3 e^{5t} .\\)\nSo the terms are:\n\\[x''' = 125e^{5t} ,\\]\n\\[- 15 x'' = (-375)e^{5t} ,\\]\n\\[75 x' = 375e^{5t} ,\\]\n\\[-125 x = -125e^{5t} .\\]\nWhen we add those terms together we get \\(0\\), and hence the differential equation we get is\n\\[x''' - 15 x'' + 75 x' - 125 x = 0 .\\]\nThe right hand side \\(0\\) is precisely what we get when we plug in \\(x=e^{5t},\\)\nand so \\(x = e^{5t}\\) is a solution to the equation.\n\u003c/p\u003e\n\u003cp\u003e\nOn the other hand when we plug in \\(x = e^x\\) to the left hand side we get\n\\[x''' - 15 x'' + 75 x' - 125 x = e^x - 15 e^x + 75 e^x - 125 e^x = (1-15+75-125) e^x \\not= 0 .\\]\nSince the left hand side is not zero when we plug in \\(x = e^x,\\)\nthen this is not a solution to the differential equation.\n\n\u003c/p\u003e\n\u003c/div\u003e","header_text":{"css":[],"js":[],"inlinejs":["\n\u003c!--\nfunction openhelpFormulas() {\nOpenWindow=window.open(\"\",\"answer_format_help\",\"width=550,height=550,status=0,toolbar=0,location=0,menubar=0,directories=0,resizeable=1,scrollbars=1\");\nOpenWindow.document.write(\"\u003ctitle\u003eHelp Entering Formulas\u003c/title\u003e\")\nOpenWindow.document.write(\"\u003cbody bgcolor='#ffffff'\u003e\")\nOpenWindow.document.write(\"\u003ccenter\u003e\u003ch2\u003eHelp Entering Formulas\u003c/h2\u003e\u003c/center\u003e\")\nOpenWindow.document.write(\"\u003cul\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003ca href='http://webwork.maa.org/wiki/Available_Functions' target='_new'\u003eLink to a list of all available functions\u003c/a\u003e\u003cbr /\u003e\u003cbr /\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eFormulas must use the correct variable(s):\u003c/font\u003e\u003cblockquote\u003eFor example, a function of time \u0026nbsp; \u003ccode\u003et\u003c/code\u003e \u0026nbsp; could be \u0026nbsp; \u003ccode\u003e-16t^2 + 12\u003c/code\u003e, while \u0026nbsp; \u003ccode\u003e-16x^2 + 12\u003c/code\u003e \u0026nbsp; would be incorrect. \u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eExamples of valid formulas:\u003c/font\u003e\u003cblockquote\u003e\u003ccode\u003e5*sin((pi*x)/2)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e5 sin(pi x/2)\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003ee^(-x)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003ee**(-x)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e1/(e^x)\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003eabs(5y)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e|5y|\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003esqrt(9 - z^2)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e(9 - z^2)^(1/2)\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003e24\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e4!\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e4 * 3 * 2 * 1\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003epi\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e4 arctan(1)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e4 atan(1)\u003c/code\u003e \u0026nbsp;\u0026nbsp;or\u0026nbsp;\u0026nbsp; \u003ccode\u003e4 tan^(-1)(1)\u003c/code\u003e\u003cbr /\u003e\u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eEntering logarithms:\u003c/font\u003e \u003cblockquote\u003e\u003cfont color='#ff0000'\u003e\u003cb\u003eCaution:\u003c/b\u003e\u003c/font\u003e In this question, use \u003ccode\u003eln(x)\u003c/code\u003e or \u003ccode\u003elog(x)\u003c/code\u003e for natural log, and \u003ccode\u003elogten(x)\u003c/code\u003e or \u003ccode\u003elog10(x)\u003c/code\u003e for the base 10 logarithm.  Enter log base b as \u003ccode\u003eln(x)/ln(b)\u003c/code\u003e\u003c/blockquote\u003e\u003cbr /\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eExamples of constants used in formulas:\u003c/font\u003e\u003cblockquote\u003e\u003ccode\u003epi\u003c/code\u003e, \u003ccode\u003ee = e^1\u003c/code\u003e\u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eExamples of operations used in formulas:\u003c/font\u003e\u003cblockquote\u003eAddition \u003ccode\u003e+\u003c/code\u003e, subtraction \u003ccode\u003e-\u003c/code\u003e, multiplication \u003ccode\u003e*\u003c/code\u003e, division \u003ccode\u003e/\u003c/code\u003e, exponentiation \u003ccode\u003e^\u003c/code\u003e (or \u003ccode\u003e**\u003c/code\u003e), factorial \u003ccode\u003e!\u003c/code\u003e\u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eExamples of functions used in formulas:\u003c/font\u003e\u003cblockquote\u003e\u003ccode\u003esqrt(x) = x^(1/2)\u003c/code\u003e, \u003ccode\u003eabs(x) = | x |\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003e2^x, e^x, ln(x), log10(x)\u003c/code\u003e \u003cbr /\u003e\u003ccode\u003esin(x), cos(x), tan(x), csc(x), sec(x), cot(x)\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003earcsin(x) = asin(x) = sin^(-1)(x)\u003c/code\u003e\u003cbr /\u003e \u003ccode\u003earccos(x) = acos(x) = cos^(-1)(x)\u003c/code\u003e\u003cbr /\u003e\u003ccode\u003earctan(x) = atan(x) = tan^(-1)(x)\u003c/code\u003e\u003cbr /\u003e\u003c/blockquote\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eSometimes formulas must be simplified:\u003c/font\u003e\u003cblockquote\u003eFor example, \u0026nbsp; \u003ccode\u003e6x + 5 - 2x\u003c/code\u003e \u0026nbsp; should be simplified to \u0026nbsp; \u003ccode\u003e4x + 5\u003c/code\u003e\u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eSometimes, certain operations are not allowed.\u003c/font\u003e\u003cblockquote\u003eUsually, the operations that are not allowed include addition \u003ccode\u003e+\u003c/code\u003e, subtraction \u003ccode\u003e-\u003c/code\u003e, multiplication \u003ccode\u003e*\u003c/code\u003e, division \u003ccode\u003e/\u003c/code\u003e, and exponentiation \u003ccode\u003e^\u003c/code\u003e (or \u003ccode\u003e**\u003c/code\u003e).  When these operations are not allowed, it is usually because you are expected to be able to simplify your answer, often without using a calculator.\u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003cli\u003e\u003cfont color='#222255'\u003eSometimes, certain functions are not allowed.\u003c/font\u003e  \u003cblockquote\u003eUsually, the functions that are not allowed include square root \u003ccode\u003esqrt( )\u003c/code\u003e, absolute value \u003ccode\u003e| |\u003c/code\u003e (or \u003ccode\u003eabs( )\u003c/code\u003e), as well as other named functions such as \u003ccode\u003esin( )\u003c/code\u003e, \u003ccode\u003eln( )\u003c/code\u003e, etc.  When these functions are not allowed, it is usually because you are expected to be able to simplify your answer, often without using a calculator.\u003c/blockquote\u003e\u003c/li\u003e\")\nOpenWindow.document.write(\"\u003c/ul\u003e\")\nOpenWindow.document.write(\"\u003c/body\u003e\")\nOpenWindow.document.write(\"\u003c/html\u003e\")\nOpenWindow.document.close()\nself.name=\"main\"\nif (window.focus) {OpenWindow.focus()}\nreturn false;\n}\n--\u003e\n"]}}],"numeric_tolerance":null,"partial_credit":null,"max_attempts":null,"allow_practice":null,"instructional":false,"sketchable":true,"print_num_spacing_lines":3,"print_page_break":false,"readonly":{"created_at":"2021-08-16T12:24:56.393Z","updated_at":"2021-08-16T12:24:56.393Z","version":11,"user_id":"58ecdcd48ccf1800120c8ac3","name":"Algorithmic (WebWorK)","skill_ids":["5a141faeac370c0004c77434"],"current_digest":"SeFvRgG6oilK2Kpio7oNsdailBbtWpx4Ak2CH5ZJFyo=","tags":"differential,equation,ode,ordinary,verify","points":7.0,"type":"webwork","multipart":true,"shallow":true,"static":true,"sources":[],"owned":false,"url":"/problems/611a591875b85800033de89e","searchable_text":"Suppose x x PARFind the value of the expression expr in terms of the variable t BITALIC Enter the terms in the order given EITALIC PAR ans rule 10 ans rule 10 ans rule 10 ans rule 10 AnswerFormatHelp formulas PARSimplify your answer above to obtaina differential equation in termsof the dependent variable x satisfiedby x x PAR expr ans rule 10 AnswerFormatHelp formulas PARSo is x x a solution to yourdifferential equation above answer 5 menu PARNow is x e t a solution to yourdifferential equation above answer 6 menu BITALIC Be sure you canjustify your answer EITALIC","abilities":["read"]},"content_license":{"license_type":"OPL","copyright_year":null,"copyright_holder":null,"source_url":"https://github.com/Edfinity/webwork-open-problem-library/blob/1a0df955ab8d9af42343ef254be8f65bc9ced13c/diffyqs-webwork/sec_0.2/prob01.pg","content_modified":null}},"readonly":{"shallow":true}}},"readonly":{"level_range":"13","created_at":"2019-03-07T22:23:15.860Z","updated_at":"2024-09-01T18:42:09.205Z","tags":"math,oer,differential_equations","author_ids":[],"licensed":true,"thumbnail":"https://d1q29jrdo98n6g.cloudfront.net/uploads/product/image/5c8199d3b8497d000d4e5eb1/thumb_notes-on-diffyq-3.jpg","free":false,"user_price":29.0,"term_price":29.0,"displayed_price":29.0,"levels":"13","author_role_title":"Contributors","abilities":[],"section":{"id":"5dd05e13c5b59e00046bd771","name":"Notes on Diffy Qs v6"}},"image":{"url":"https://d1q29jrdo98n6g.cloudfront.net/uploads/product/image/5c8199d3b8497d000d4e5eb1/notes-on-diffyq-3.jpg","thumb_url":"https://d1q29jrdo98n6g.cloudfront.net/uploads/product/image/5c8199d3b8497d000d4e5eb1/thumb_notes-on-diffyq-3.jpg"},"related_products":[],"toc":{"assignments":[{"id":"5dd05f39c5b59e00046be3c2","name":"Edfinity Demo"},{"id":"5dd05e13c5b59e00046bd779","name":"Sec 0.2:  Introduction to differential equations"},{"id":"5dd05e19c5b59e00046bd7dc","name":"Sec 0.3:  Classification of differential equations"},{"id":"5dd05e1ec5b59e00046bd849","name":"Sec 1.1:  Integrals as solutions"},{"id":"5dd05e22c5b59e00046bd88b","name":"Sec 1.2:  Slope fields"},{"id":"5dd05e27c5b59e00046bd8be","name":"Sec 1.3:  Separable equations"},{"id":"5dd05e2bc5b59e00046bd8ea","name":"Sec 1.4:  Linear equations and the integrating factor"},{"id":"5dd05e30c5b59e00046bd937","name":"Sec 1.5:  Substitution"},{"id":"5dd05e35c5b59e00046bd95e","name":"Sec 1.6:  Autonomous equations"},{"id":"5dd05e3cc5b59e00046bd99d","name":"Sec 1.7:  Numerical methods"},{"id":"5dd05e3ec5b59e00046bd9c3","name":"Sec 1.8:  Exact equations"},{"id":"5dd05e3fc5b59e00046bd9cd","name":"Sec 1.9: First order linear PDE"},{"id":"5dd05e43c5b59e00046bda27","name":"Sec 2.1:  Second order linear ODEs"},{"id":"5dd05e46c5b59e00046bda58","name":"Sec 2.2:  Constant coefficient second order linear ODEs"},{"id":"5dd05e4bc5b59e00046bda90","name":"Sec 2.3:  Higher order linear ODEs"},{"id":"5dd05e4fc5b59e00046bdacd","name":"Sec 2.4:  Mechanical vibrations"},{"id":"5dd05e53c5b59e00046bdafe","name":"Sec 2.5:  Nonhomogeneous equations"},{"id":"5dd05e53c5b59e00046bdb07","name":"Sec 2.6:  Forced oscillations and resonance"},{"id":"5dd05e55c5b59e00046bdb22","name":"Sec 3.1:  Introduction to systems of ODEs"},{"id":"5dd05e56c5b59e00046bdb38","name":"Sec 3.2:  Matrices and linear systems"},{"id":"5dd05e59c5b59e00046bdb78","name":"Sec 3.3:  Linear systems of ODEs"},{"id":"5dd05e5bc5b59e00046bdb86","name":"Sec 3.4:  Eigenvalue method"},{"id":"5dd05e5dc5b59e00046bdb9f","name":"Sec 3.5:  Two dimensional systems and their vector fields"},{"id":"5dd05e64c5b59e00046bdbd6","name":"Sec 3.6:  Second order systems and applications"},{"id":"5dd05e68c5b59e00046bdc27","name":"Sec 3.7:  Multiple eigenvalues"},{"id":"5dd05e6bc5b59e00046bdc5d","name":"Sec 3.8:  Matrix exponentials"},{"id":"5dd05e6ec5b59e00046bdc99","name":"Sec 3.9:  Nonhomogeneous systems"},{"id":"5dd05e70c5b59e00046bdcbd","name":"Sec 4.10:  Dirichlet problem in the circle and the Poisson kernel"},{"id":"5dd05e70c5b59e00046bdcb3","name":"Sec 4.1:  Boundary value problems"},{"id":"5dd05e72c5b59e00046bdce2","name":"Sec 4.2:  The trigonometric series"},{"id":"5dd05e74c5b59e00046bdcf3","name":"Sec 4.3:  More on the Fourier series"},{"id":"5dd05e76c5b59e00046bdd10","name":"Sec 4.4:  Sine and cosine series"},{"id":"5dd05e7ac5b59e00046bdd4a","name":"Sec 4.5:  Applications of Fourier series"},{"id":"5dd05e7dc5b59e00046bdd60","name":"Sec 4.6:  PDEs, separation of variables, and the heat equation"},{"id":"5dd05e7fc5b59e00046bdd85","name":"Sec 4.7:  One dimensional wave equation"},{"id":"5dd05e83c5b59e00046bdde0","name":"Sec 4.8:  D’Alembert solution of the wave equation"},{"id":"5dd05e86c5b59e00046bde12","name":"Sec 4.9:  Steady state temperature and the Laplacian"},{"id":"5dd05e89c5b59e00046bde42","name":"Sec 5.1:  Sturm-Liouville problems"},{"id":"5dd05e8ac5b59e00046bde47","name":"Sec 6.1:  The Laplace transform"},{"id":"5dd05e8cc5b59e00046bde56","name":"Sec 6.2:  Transforms of derivatives and ODEs"},{"id":"5dd05e8ec5b59e00046bde68","name":"Sec 6.3:  Convolution"},{"id":"5dd05e8fc5b59e00046bde71","name":"Sec 6.4:  Dirac delta and impulse response"},{"id":"5dd05e90c5b59e00046bde7f","name":"Sec 6.5: Solving PDEs with the Laplace transform"},{"id":"5dd05e91c5b59e00046bde8f","name":"Sec 7.1:  Power series"},{"id":"5dd05e93c5b59e00046bdea3","name":"Sec 7.2:  Series solutions of linear second order ODEs"},{"id":"5dd05e94c5b59e00046bdeb8","name":"Sec 7.3:  Singular points and the method of Frobenius"},{"id":"5dd05e95c5b59e00046bdec9","name":"Sec 8.1:  Linearization, critical points, and equilibria"},{"id":"5dd05e99c5b59e00046bdef0","name":"Sec 8.2:  Stability and classification of isolated critical points"},{"id":"5dd05e9ac5b59e00046bdef8","name":"Sec 8.3:  Applications of nonlinear systems"},{"id":"5dd05e9bc5b59e00046bdf01","name":"Sec 8.4:  Limit cycles"},{"id":"5dd05e9bc5b59e00046bdf06","name":"Sec A.1:  Vectors, mappings, and matrices"},{"id":"5dd05e9fc5b59e00046bdfa8","name":"Sec A.2: Matrix algebra"},{"id":"5dd05ea6c5b59e00046be05a","name":"Sec A.3: Elimination"},{"id":"5dd05eadc5b59e00046be180","name":"Sec A.4:  Subspaces, dimension, and the kernel"},{"id":"5dd05eb4c5b59e00046be2a4","name":"Sec A.5: Inner product and projections"},{"id":"5dd05eb8c5b59e00046be364","name":"Sec A.6:  Determinant"}]}}