{"id":24781,"date":"2022-01-03T19:59:18","date_gmt":"2022-01-03T14:29:18","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=24781"},"modified":"2024-07-11T18:12:43","modified_gmt":"2024-07-11T12:42:43","slug":"successive-differentiation-excercise","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=24781","title":{"rendered":"SUCCESSIVE DIFFERENTIATION (Excercise)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1:}\\ if\\ y\\ =\\ x^2\\ +\\ 6\\ x\\ -\\ 15\\ \\color {red} {Find\\ \\frac{d^2y}{dx^2}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ x^2\\ +\\ 6\\ x\\ -\\ 15\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d}{dx}(y)\\ =\\ \\frac{d}{dx}( x^2)\\  +\\ 6\\ \\frac{d}{dx}( x) -\\ \\frac{d}{dx}(15)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 2\\ x\\ +\\ 6\\ (1)\\ -\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 2\\ x\\ +\\ 6\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Again\\ Differentiate\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d}{dx}(\\frac{dy}{dx})\\ =\\ \\frac{d}{dx}( 2\\ x\\ +\\ 6)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d^2y}{dx^2}\\ =\\ 2\\  \\frac{d}{dx}(x)\\ +\\ \\frac{d}{dx}(6)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d^2y}{dx^2}\\ =\\  2\\ (1)\\  +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d^2y}{dx^2}\\ =\\  2\\    \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- 350 x 250 ad -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:350px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3583972194\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3:}\\  If\\ y\\ =\\ x^2\\ cos\\  x,\\ \\color {red} {prove\\ that\\ x^2\\ y_2\\ -\\ 4\\ x\\ y_1\\ +\\ (x^2\\ +\\ 6)y}\\ =\\ 0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\  x^2\\ cos\\  x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d}{dx}(y)\\ =\\ \\frac{d}{dx}( x^2\\ cos\\  x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x^2\\ \\frac{d}{dx}(cos\\ x)\\ +\\ cos\\ x\\ \\frac{d}{dx}(x^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_1\\ =\\ x^2\\ (-\\ sin\\ x)\\ +\\ cos\\ x\\ 2\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_1\\ =\\ -\\ x^2\\ sin\\ x\\ +\\ 2\\ x\\ cos\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Again\\ Differentiate\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d}{dx}(y_1)\\ =\\ \\frac{d}{dx}(-\\ x^2\\ sin\\ x\\ +\\ 2\\ x\\ cos\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_2\\ =\\ -\\ \\frac{d}{dx}(x^2\\ sin\\ x)\\ +\\ 2\\ \\frac{d}{dx}(x\\ cos\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_2\\ =\\ -\\ (x^2\\ \\frac{d}{dx}(sin\\ x)\\  +\\  sin\\ x\\ \\frac{d}{dx}(x^2))\\ +\\ 2[x\\ \\frac{d}{dx}(cos\\ x)\\ +\\ cos\\ x\\ \\frac{d}{dx}(x)]\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_2\\ =\\ -( x^2\\ cos\\ x\\ +\\  sin\\ x\\ (2\\ x))\\ +\\ 2[x\\ (-\\ sin\\ x)\\ +\\ cos\\ x\\ (1)]\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_2\\ =\\ -\\ x^2\\ cos\\ x\\ -\\  2\\ x\\ sin\\ x\\ -\\ 2 x\\ sin\\ x\\ +\\ 2\\ cos\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y_2\\ =\\ -\\ x^2\\ cos\\ x\\ -\\  4\\ x\\ sin\\ x\\ +\\ 2\\ cos\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\  x^2\\ y_2\\ -\\ 4\\ x\\ y_1\\ +\\ (x^2\\ +\\ 6)y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ x^2\\ (-\\ x^2\\ cos\\ x\\ -\\  4\\ x\\ sin\\ x\\ +\\ 2\\ cos\\ x)\\ -\\ 4\\ x\\ (-\\ x^2\\ sin\\ x\\ +\\ 2\\ x\\ cos\\ x)\\ +\\ (x^2\\ +\\ 6)(x^2\\ cos\\  x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ x^4\\ cos\\ x\\ -\\  4\\ x^3\\ sin\\ x\\ +\\ 2\\ x^2\\ cos\\ x\\ +\\ 4\\ x^3\\ sin\\ x\\ -\\ 8\\ x^2\\ cos\\ x\\ +\\ x^4\\ cos\\ x+\\ 6\\ x^2\\ cos\\  x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\ =\\ R.\\ H.\\ S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4:}\\  If\\ xy\\ =\\ a\\ e^x\\ +\\ b\\ e^{-x}\\ \\color {red} {prove\\ that\\ xy_2\\ +\\ 2\\ y_1\\ =\\ xy}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  xy\\ =\\ a\\ e^x\\ +\\ b\\ e^{-x}\\ &#8212;&#8211;\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d}{dx}(xy)\\ =\\ a\\ \\frac{d}{dx}( e^x)\\ +\\ b\\ \\frac{d}{dx}( e^{-x}) \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{d}{dx}(y)\\ +\\ y\\  \\frac{d}{dx}(x)\\ =\\ a\\ e^x\\  +\\ b\\ e^{-x} (-1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[xy_1\\ +\\ y\\  (1)\\ =\\ a\\ e^x\\  -\\ b\\ e^{-x}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[xy_1\\ +\\ y\\ =\\ a\\ e^x\\  -\\ b\\ e^{-x}\\ &#8212;-\\ (2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Again\\ Differentiate\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{d}{dx}(x\\ y_1)\\ +\\ \\frac{d}{dx}(y)\\ =\\  a\\ \\frac{d}{dx}(e^x)\\ -\\ b\\ \\frac {d}{dx}(e^{-x})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{d}{dx}(y_1)\\ +\\ y_1\\  \\frac{d}{dx}(x) +\\ y_1\\ =\\ a\\ e^x\\  -\\ b\\ e^{-x} (-\\ 1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[xy_2\\ +\\ y_1\\  (1)\\ +\\ y_1 =\\ a\\ e^x\\  +\\ b\\ e^{-x}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[xy_2\\ +\\ 2\\ y_1\\ =\\ xy\\  &#8212;&#8212;-\\ Using\\ (1)\\ 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