Example 20 - Chapter 9 Class 12 Differential Equations

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Example 20 Find the particular solution of the differential equation log(š‘‘š‘¦/š‘‘š‘„)=3š‘„+4š‘¦ given that š‘¦=0 š‘¤ā„Žš‘’š‘› š‘„=0 log(š‘‘š‘¦/š‘‘š‘„)=3š‘„+4š‘¦ š’…š’š/š’…š’™ = e(3x + 4y) š‘‘š‘¦/š‘‘š‘„ = e3x e4y Separating the variables š‘‘š‘¦/š‘’^4š‘¦ = e3x dx eāˆ’4y dy = e3x dx Integrating both sides āˆ«1ā–’怖š‘’^(āˆ’4š‘¦) š‘‘š‘¦ć€—=āˆ«1ā–’š‘’^3š‘„ š‘‘š‘„ š’†^(āˆ’šŸ’š’š)/(āˆ’šŸ’)=š’†^šŸ‘š’™/šŸ‘+š‘Ŗ 0=š‘’^3š‘„/3+š‘’^(āˆ’4š‘¦)/4+š¶ š‘’^3š‘„/3 + š‘’^(āˆ’4š‘¦)/4 + C = 0 (4š‘’3š‘„" + " 3š‘’^(āˆ’4š‘¦))/12 + C = 0 (4š‘’3š‘„" + " 3š‘’^(āˆ’4š‘¦) + 12š¶)/12 " = 0" šŸ’š’†^šŸ‘š’™ + šŸ‘š’†^(āˆ’šŸ’š’š) + 12C = 0 Given y = 0 when x = 0 Putting x = 0 & y = 0 in (1) 4š‘’^(3(0)) + 3š‘’^(āˆ’4(0)) + 12C = 0 4š‘’^0 + 3š‘’^0 + 12C = 0 4 + 3 + 12C = 0 7 + 12C = 0 12C = ā€“ 7 C = (āˆ’šŸ•)/šŸšŸ Putting value of C in (1) 4š‘’^3š‘„ + 3š‘’^(āˆ’4š‘¦) + 12C = 0 4š‘’^3š‘„ + 3š‘’^(āˆ’4š‘¦) + 12((āˆ’7)/12) = 0 šŸ’š’†^šŸ‘š’™ + šŸ‘š’†^(āˆ’šŸ’š’š) āˆ’ 7 = 0

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo