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Last updated at Dec. 11, 2019 by Teachoo
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Ex 9.4, 5 For each of the differential equations in Exercises 1 to 10, find the general solution : (๐^๐ฅ+๐^(โ๐ฅ) )๐๐ฆโ(๐^๐ฅโ๐^(โ๐ฅ) )๐๐ฅ=0 (๐^๐ฅ+๐^(โ๐ฅ) )๐๐ฆโ(๐^๐ฅโ๐^(โ๐ฅ) )๐๐ฅ=0 (๐^๐ฅ+๐^(โ๐ฅ) )๐๐ฆ = (๐^๐ฅโ๐^(โ๐ฅ) )๐๐ฅ ๐๐ฆ/๐๐ฅ = (๐^๐ฅ โ ๐^(โ๐ฅ))/(๐^๐ฅ + ๐^(โ๐ฅ) ) dx ๐๐ฆ = (๐^๐ฅ โ ๐^(โ๐ฅ))/(๐^๐ฅ + ๐^(โ๐ฅ) ) dx Integrating both sides. โซ1โ๐๐ฆ = โซ1โ(๐^๐ฅ โ ๐^(โ๐ฅ))/(๐^๐ฅ + ๐^(โ๐ฅ) ) dx ๐ฆ = โซ1โ(๐^๐ฅ โ ๐^(โ๐ฅ))/(๐^๐ฅ + ๐^(โ๐ฅ) ) dx Let t = ๐^๐ฅ+๐^(โ๐ฅ) ๐๐ก/๐๐ฅ = (๐^๐ฅโ๐^(โ๐ฅ) ) dx = ๐๐ก/(๐^๐ฅ โ ๐^(โ๐ฅ) ) Putting value of t and dt in (1) โซ1โ๐๐ฆ = โซ1โ(๐^(๐ฅ )โใ ๐ใ^(โ๐ฅ))/๐ก ๐๐ก/(๐^๐ฅ โ ๐^(โ๐ฅ) ) . โซ1โ๐๐ฆ = โซ1โใ๐๐ก/๐ก " " ใ y = log |๐ก|+๐ Putting back t = ๐^๐ฅโ๐^(โ๐ฅ) y = log |๐^๐ฅโ๐^(โ๐ฅ) | + C y = log (๐^๐โ๐^(โ๐)) + C As ๐^๐ฅโ๐^(โ๐ฅ) > 0 So, its always positive Removing the modulus
Ex 9.4
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Ex 9.4, 5 You are here
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