{"id":304025,"date":"2022-09-10T22:48:48","date_gmt":"2022-09-10T17:18:48","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/if-a-function-yfx-satisfies-the-differential-equation-fx%e2%8b%85sin%e2%81%a12x%e2%88%92cos%e2%81%a1x1sin2%e2%81%a1xf%e2%80%b2x0-with-initial-condition-y00\/"},"modified":"2022-09-10T22:48:48","modified_gmt":"2022-09-10T17:18:48","slug":"if-a-function-yfx-satisfies-the-differential-equation-fx%e2%8b%85sin%e2%81%a12x%e2%88%92cos%e2%81%a1x1sin2%e2%81%a1xf%e2%80%b2x0-with-initial-condition-y00","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/if-a-functionyfxsatisfies-the-differential-equationfx\/","title":{"rendered":"\u00a0If a function\u00a0y=f(x)\u00a0satisfies the differential equation\u00a0f(x)\u22c5sin\u20612x\u2212cos\u2061x+1+sin2\u2061xf\u2032(x)=0\u00a0with initial condition\u00a0y(0)=0,\u00a0then the value\u00a0\u00a0of\u00a0f\u03c06\u00a0is equal to\u00a0"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"\u00a0If a function\u00a0y=f(x)\u00a0satisfies the differential equation\u00a0f(x)\u22c5sin\u20612x\u2212cos\u2061x+1+sin2\u2061xf\u2032(x)=0\u00a0with initial condition\u00a0y(0)=0,\u00a0then the value\u00a0\u00a0of\u00a0f\u03c06\u00a0is equal to\u00a0","custom_permalink":"question\/mathematics\/if-a-functionyfxsatisfies-the-differential-equationfx\/"},"categories":[],"acf":{"question":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtext>&nbsp;If a function&nbsp;<\/mtext><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mtext>&nbsp;satisfies the differential equation&nbsp;<\/mtext><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>\u22c5<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><msup><mi>f<\/mi><mi mathvariant=\"normal\">\u2032<\/mi><\/msup><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><mn>0<\/mn><mtext>&nbsp;with initial condition&nbsp;<\/mtext><mi>y<\/mi><mo>(<\/mo><mn>0<\/mn><mo>)<\/mo><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mtext>&nbsp;then the value&nbsp;<\/mtext><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtext>&nbsp;of&nbsp;<\/mtext><mi>f<\/mi><mfenced separators=\"|\"><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mfenced><mtext>&nbsp;is equal to&nbsp;<\/mtext><\/math><\/p>","options":[{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>1<\/mn><mn>5<\/mn><\/mfrac><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>4<\/mn><mn>5<\/mn><\/mfrac><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>2<\/mn><mn>5<\/mn><\/mfrac><\/math><\/p>","correct":true}],"solution":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mtext>&nbsp;The given equation is&nbsp;<\/mtext><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>\u22c5<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><msup><mi>f<\/mi><mi mathvariant=\"normal\">\u2032<\/mi><\/msup><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>y<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfenced separators=\"|\"><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/mfenced><mi>y<\/mi><mo>=<\/mo><mfrac><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mtext>&nbsp;is in the form of&nbsp;<\/mtext><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>P<\/mi><mi>y<\/mi><mo>=<\/mo><mi>Q<\/mi><\/mtd><\/mtr><mtr><mtd><mtext>&nbsp;Where&nbsp;<\/mtext><mi>P<\/mi><mo>=<\/mo><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mtext>&nbsp;and&nbsp;<\/mtext><mi>Q<\/mi><mo>=<\/mo><mfrac><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mtext>&nbsp;Integrating Factor&nbsp;<\/mtext><mo>(<\/mo><mi>IF<\/mi><mo>)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo>\u222b<\/mo><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mi>d<\/mi><mi>x<\/mi><\/mrow><\/msup><\/mtd><\/mtr><mtr><mtd><mo>=<\/mo><msup><mi>e<\/mi><mrow><mi>ln<\/mi><mo>\u2061<\/mo><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><\/mrow><\/msup><mspace width=\"1em\">&nbsp;<\/mspace><mfenced separators=\"|\"><mrow><mo>\u2235<\/mo><mo>\u222b<\/mo><mfrac><mrow><msup><mi>f<\/mi><mi mathvariant=\"normal\">\u2032<\/mi><\/msup><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><mrow><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/mfrac><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mi>ln<\/mi><mo>\u2061<\/mo><mo>|<\/mo><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>|<\/mo><mo>+<\/mo><mi mathvariant=\"normal\">C<\/mi><\/mrow><\/mfenced><\/mtd><\/mtr><mtr><mtd><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mtd><\/mtr><mtr><mtd><mtext>&nbsp;General solution is&nbsp;<\/mtext><mi>y<\/mi><mo>\u22c5<\/mo><mo>(<\/mo><mi>IF<\/mi><mo>)<\/mo><mo>=<\/mo><mo>\u222b<\/mo><mo>(<\/mo><mi>IF<\/mi><mo>)<\/mo><mi>Q<\/mi><mi>d<\/mi><mi>x<\/mi><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>y<\/mi><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><mo>=<\/mo><mo>\u222b<\/mo><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><mfrac><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><\/mfrac><mi>d<\/mi><mi>x<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" rowalign=\"baseline baseline baseline baseline baseline top\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>y<\/mi><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><\/mfenced><mo>=<\/mo><mo>\u222b<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><\/mtd><\/mtr><mtr><mtd><mtext>&nbsp;When&nbsp;<\/mtext><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><mo stretchy=\"false\">\u21d2<\/mo><mi>C<\/mi><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mtext>&nbsp;When&nbsp;<\/mtext><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><mo>,<\/mo><mtext>&nbsp;then&nbsp;<\/mtext><mi>y<\/mi><mfenced separators=\"|\"><mrow><mn>1<\/mn><mo>+<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><\/mfenced><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>y<\/mi><mfenced separators=\"|\"><mfrac><mn>5<\/mn><mn>4<\/mn><\/mfrac><\/mfenced><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mfrac><mn>2<\/mn><mn>5<\/mn><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo>\u2234<\/mo><mi>f<\/mi><mfenced separators=\"|\"><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mfenced><mo>=<\/mo><mfrac><mn>2<\/mn><mn>5<\/mn><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/p><p>Therefore, the correct answer is (4).<\/p>","subject":""},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u00a0If a function\u00a0y=f(x)\u00a0satisfies the differential equation\u00a0f(x)\u22c5sin\u20612x\u2212cos\u2061x+1+sin2\u2061xf\u2032(x)=0\u00a0with initial condition\u00a0y(0)=0,\u00a0then the value\u00a0\u00a0of\u00a0f\u03c06\u00a0is equal to\u00a0 - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"\u00a0If a function\u00a0y=f(x)\u00a0satisfies the differential equation\u00a0f(x)\u22c5sin\u20612x\u2212cos\u2061x+1+sin2\u2061xf\u2032(x)=0\u00a0with initial condition\u00a0y(0)=0,\u00a0then the value\u00a0\u00a0of\u00a0f\u03c06\u00a0is equal to\u00a0\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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