Variable separation - Equation given
Example 20
Misc 13 (MCQ)
Ex 9.3, 3
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Ex 9.3, 7 Important
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Ex 9.3, 6
Ex 9.3, 2
Misc 4
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Ex 9.3, 5
Ex 9.3, 14
Ex 9.3, 13
Example 7 Important
Ex 9.3, 16
Ex 9.3, 12
Ex 9.3, 11 Important
Ex 9.3, 1 Important
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Misc 7 Important
Ex 9.3, 4 Important
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Misc 5 Important
Variable separation - Equation given
Last updated at April 16, 2024 by Teachoo
Ex 9.3, 9 For each of the differential equations in Exercises 1 to 10, find the general solution : ๐๐ฆ/๐๐ฅ=sin^(โ1)โก๐ฅ๐๐ฆ/๐๐ฅ=sin^(โ1)โก๐ฅ ๐๐ฆ = sin^(โ1)โก๐ฅ dx Integrating both sides โซ1โใ๐ ๐ ใ= โซ1โใใ๐ฌ๐ข๐งใ^(โ๐)โกใ๐.๐ ๐ ๐ใ ใ Integrating by parts, using formula โซ1โใ๐ (๐ฅ)๐(๐ฅ)๐๐ฅ ใ= ๐(๐ฅ) โซ1โใ๐(๐ฅ)๐๐ฅ โโซ1โใ[๐โฒ(๐ฅ)โซ1โ๐(๐ฅ)๐๐ฅ] ๐๐ฅ ใ ใ Take f(x) = sinโ1 x and g(x) = 1 y = x ใ๐๐๐ใ^(โ๐) ๐ โ โซ1โ๐/โ(๐ โ ๐^๐ ) dx Let t = 1 โ x2 dt = โ2xdx x dx = (โ๐๐ก)/2 Hence, our equation becomes y = x sinโ1 x โ โซ1โ(โ๐๐ก)/(2โ๐ก) y = x sinโ1 x + โซ1โ๐๐ก/(2โ๐ก) y = x sinโ1 x + ๐/๐ โซ1โใ๐^((โ๐)/๐) ๐ ๐ใ y = x sinโ1 x + ๐/๐ ๐^((โ๐)/๐ + ๐)/((โ๐)/๐ + ๐) + C y = x sinโ1 x + 1/2 (๐ก^(1/2) )/((1/2) )+๐ถ y = x sinโ1 x + โ๐ก + C Putting back value of t y = x sinโ1 x + โ(๐โ๐^๐ ) + C y = sinโ1 x โซ1โใ๐ ๐ ๐ โโซ1โ[๐/โ(๐ โ ๐^๐ ) โซ1โใ๐.๐ ๐ ใ] ใ dx