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Ex 9.4, 16 - For xy dy/dx = (x + 2) (y + 2), find solution - Variable separation - Equation given

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.4, 16 For the differential equation ๐‘ฅ๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=(๐‘ฅ+2)(๐‘ฆ+2) , find the solution curve passing through the point (1 , โˆ’1) ๐‘ฅ๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=(๐‘ฅ+2)(๐‘ฆ+2) (๐‘ฆ ๐‘‘๐‘ฆ)/(๐‘ฆ + 2) = (๐‘ฅ + 2)/๐‘ฅ dx Integrating both sides โˆซ1โ–’๐‘ฆ/(๐‘ฆ + 2) dy = โˆซ1โ–’(๐‘ฅ + 2)/๐‘ฅ dx โˆซ1โ–’(๐‘ฆ + 2 โˆ’ 2)/(๐‘ฆ + 2) dy = โˆซ1โ–’(1+( 2)/๐‘ฅ) ๐‘‘๐‘ฅ โˆซ1โ–’(1โˆ’( 2)/(๐‘ฆ + 2)) dy = โˆซ1โ–’(1+( 2)/๐‘ฅ) ๐‘‘๐‘ฅ y โˆ’ 2 log (y + 2) = x + 2 log x + C Since curve passes through (1, โˆ’ 1) Putting x = 1 and y = โˆ’1 in (1) โˆ’1 โˆ’ 2 log (โˆ’1 + 2) = 1 + 2 log 1 + C โˆ’1 โˆ’ 2log1 = 1 + 2log1 + C โˆ’1 = 1 + C C = โˆ’2 Put C = โˆ’2 In (1) y = 2 log (y + 2) + x + 2 log x โˆ’ 2 y โˆ’ x + 2 = log (y + 2)2 + log x2 y โˆ’ x + 2 = log (x2 (y + 2)2)

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