{"id":1440,"date":"2020-03-26T06:09:05","date_gmt":"2020-03-26T06:09:05","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1440"},"modified":"2020-03-26T10:52:04","modified_gmt":"2020-03-26T10:52:04","slug":"ibdp-past-year-exam-questions-differential-equations","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/differential-equations\/ibdp-past-year-exam-questions-differential-equations\/","title":{"rendered":"IBDP Past Year Exam Questions &#8211; Differential Equations"},"content":{"rendered":"<p><strong>1.\u00a0 \u00a0[M09\/P3\/TZ0]<\/strong><\/p>\r\n<p>Consider the differential equation\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>d<\/mo><mi>y<\/mi><\/mrow><mrow><mo>d<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>&#160;<\/mo><mo>+<\/mo><mo>&#160;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/math>\r\n for which \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mo>&#8211;<\/mo><mn>1<\/mn><\/math>\r\n when \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/math>\r\n .<\/p>\r\n<p>(a)\u00a0 \u00a0Use Euler&#8217;s method with a step length of 0.25 to find an estimate for the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math>\r\n when\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/math>\r\n .\u00a0 \u00a0[7 marks]<\/p>\r\n<p>(b)\u00a0 \u00a0(i)\u00a0 \u00a0Solve the differential equation giving your answer in the form \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><\/math>\r\n .<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 (ii)\u00a0 Find the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math>\r\n when\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/math>\r\n .\u00a0 \u00a0[13 marks]<\/p>\r\n<div id=\"link1-link-1440\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1440, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1440\">Solution<\/span><\/a><\/div><div id=\"link1-content-1440\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1446\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-16.jpg\" alt=\"\" width=\"714\" height=\"942\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-16.jpg 714w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-16-227x300.jpg 227w\" sizes=\"auto, (max-width: 714px) 100vw, 714px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>2.\u00a0 \u00a0[N09\/P3\/TZ0]<\/strong><\/p>\r\n<p>Solve the differential equation\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>d<\/mo><mi>y<\/mi><\/mrow><mrow><mo>d<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mi>y<\/mi><mi>x<\/mi><\/mfrac><mo>&#160;<\/mo><mo>+<\/mo><mo>&#160;<\/mo><mfrac><msup><mi>y<\/mi><mn>2<\/mn><\/msup><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo>&#160;<\/mo><\/math>\r\n (where\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>&#62;<\/mo><mn>0<\/mn><\/math>\r\n )<\/p>\r\n<p>given that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/math>\r\n when\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/math>\r\n . give your answer in the form\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><\/math>\r\n .\u00a0 \u00a0[13 marks]<\/p>\r\n<div id=\"link2-link-1440\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1440, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1440\">Solution<\/span><\/a><\/div><div id=\"link2-content-1440\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1449\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-10.jpg\" alt=\"\" width=\"970\" height=\"758\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-10.jpg 970w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-10-300x234.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-10-768x600.jpg 768w\" sizes=\"auto, (max-width: 970px) 100vw, 970px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>3. [M10\/P3\/TZ0]<\/strong><\/p>\r\n<p>Given that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>d<\/mo><mi>y<\/mi><\/mrow><mrow><mo>d<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mo>&#8211;<\/mo><mn>2<\/mn><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>e<\/mi><mi>x<\/mi><\/msup><\/math>\r\n \u00a0and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/math>\r\n when\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/math>\r\n , use Euler\u2019s method with a step length of 0.1 to find an approximation for the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math>\r\n when\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><mo>.<\/mo><mn>4<\/mn><\/math>\r\n . Give all intermediate values with maximum possible accuracy.<\/p>\r\n<div id=\"link3-link-1440\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1440, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1440\">Solution<\/span><\/a><\/div><div id=\"link3-content-1440\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1458\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-8.jpg\" alt=\"\" width=\"939\" height=\"358\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-8.jpg 939w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-8-300x114.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-8-768x293.jpg 768w\" sizes=\"auto, (max-width: 939px) 100vw, 939px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>4. [N10\/P3\/TZ0]<\/strong><\/p>\r\n<p>Solve the differential equation\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced><mrow><mi>x<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><\/mfenced><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>&#160;<\/mo><mo>+<\/mo><mo>&#160;<\/mo><mi>x<\/mi><mi>y<\/mi><mo>=<\/mo><mfenced><mrow><mi>x<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><\/mfenced><msup><mi>e<\/mi><mrow><mo>&#8211;<\/mo><mi>x<\/mi><\/mrow><\/msup><\/math>\r\n<\/p>\r\n<p>given that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/math>\r\n when\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/math>\r\n . Give your answer in the form\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><\/math>\r\n .\u00a0 \u00a0 [13 marks]<\/p>\r\n<div id=\"link4-link-1440\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1440, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1440\">Solution<\/span><\/a><\/div><div id=\"link4-content-1440\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1474\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-9.jpg\" alt=\"\" width=\"933\" height=\"657\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-9.jpg 933w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-9-300x211.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-9-768x541.jpg 768w\" sizes=\"auto, (max-width: 933px) 100vw, 933px\" \/><\/p>\r\n<p>\u00a0<\/div><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"1.\u00a0 \u00a0[M09\/P3\/TZ0] Consider the differential equation\u00a0 dydx=y2&#160;+&#160;x22x2 for which y=&#8211;1 when x=1 . (a)\u00a0 \u00a0Use Euler&#8217;s method with a step length of 0.25 to find an estimate for the value of\u00a0 y when\u00a0 x=2 .\u00a0 \u00a0[7 marks] (b)\u00a0 \u00a0(i)\u00a0 \u00a0Solve the differential equation giving your answer in the form y=fx . \u00a0 \u00a0 \u00a0 \u00a0 [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1440","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-differential-equations","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1440","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1440"}],"version-history":[{"count":10,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1440\/revisions"}],"predecessor-version":[{"id":1475,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1440\/revisions\/1475"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}