# Ex 9.5, 9 - Chapter 9 Class 12 Differential Equations

Last updated at April 16, 2024 by Teachoo

Ex 9.5

Ex 9.5, 1
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Ex 9.5, 2

Ex 9.5, 3 Important

Ex 9.5, 4

Ex 9.5, 5 Important

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Ex 9.5, 7 Important

Ex 9.5, 8 Important

Ex 9.5, 9 You are here

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Ex 9.5, 12 Important

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Ex 9.5, 17 Important

Ex 9.5, 18 (MCQ)

Ex 9.5, 19 (MCQ) Important

Last updated at April 16, 2024 by Teachoo

Ex 9.5, 9 For each of the differential equation find the general solution : ๐ฅ ๐๐ฆ/๐๐ฅ+๐ฆโ๐ฅ+๐ฅ๐ฆ cotโกใ๐ฅ=0(๐ฅโ 0)ใ Given equation x ๐๐ฆ/๐๐ฅ + y โ x + xy cot x = 0 Dividing both sides by x ๐๐ฆ/๐๐ฅ + ๐ฆ/๐ฅ โ 1 + y cot x = 0 ๐๐ฆ/๐๐ฅ + y (1/๐ฅ+cotโก๐ฅ ) โ 1 = 0 ๐ ๐/๐ ๐ + (๐/๐+๐๐๐โก๐ ) y = 1 Comparing (1) with ๐๐ฆ/๐๐ฅ + Py = Q P = ๐/๐ + cot x & Q = 1 Finding integrating factor, I.F. I.F. = e^โซ1โใ๐ ๐๐ฅ ใ = e^โซ1โ(1/๐ฅ + cotโก๐ฅ )๐๐ฅ = e^โซ1โใ1/๐ฅ ๐๐ฅ + โซ1โใcotโก๐ฅ ๐๐ฅใใ = ๐^(logโก๐ฅ + logโกsinโก๐ฅ ) = ๐^logโกใ(๐ฅ sinโก๐ฅ)ใ = x sin x Solution of the equation is y ร I.F. = โซ1โใQร๐ผ๐นใโก๐๐ฅ + C y (x sin x) = โซ1โใ๐.ใ๐๐๐ ๐ใโก๐ ๐ ใ Let I = โซ1โใ๐.๐ฌ๐ข๐งโกใ๐.๐ ๐ใ ใ I = x โซ1โsinโกใ๐ฅ ๐๐ฅโโซ1โ[1.โซ1โsinโกใ๐ฅ ๐๐ฅใ ]๐๐ฅใ = x (โ cos x) โ โซ1โใ1.(โcosโกใ๐ฅ)ใ ๐๐ฅใ = โ x. cos x + โซ1โcosโกใ๐ฅ ๐๐ฅใ Using formula โซ1โใ๐(๐ฅ)๐(๐ฅ)๐๐ฅ=๐(๐ฅ)๐๐(๐ฅ)๐๐ฅโโซ1โ[๐โฒ(๐ฅ)][๐(๐ฅ)๐๐ฅ] ใ dx Taking f(x) = x & g(x) = sin x = โ x cos x + sin x Putting value of I in (2), y x sin x = โx cos x + sin x + C Divide by x sin x y = (โ๐ ๐๐๐โก๐)/(๐ ๐๐๐โก๐ ) + ๐๐๐โก๐/(๐ ๐๐๐โก๐ ) + ๐ช/(๐ ๐๐๐โก๐ ) y = โcot x + 1/๐ฅ + ๐ถ/(๐ฅ ๐ ๐๐โก๐ฅ ) y = ๐/๐ โ cot x + ๐ช/(๐ ๐๐๐โก๐ ) Which is the general solution of the given differential equation = โ x cos x + sin x Putting value of I in (2), y x sin x = โx cos x + sin x + C Divide by x sin x y = (โ๐ ๐๐๐โก๐)/(๐ ๐๐๐โก๐ ) + ๐๐๐โก๐/(๐ ๐๐๐โก๐ ) + ๐ช/(๐ ๐๐๐โก๐ ) y = โcot x + 1/๐ฅ + ๐ถ/(๐ฅ ๐ ๐๐โก๐ฅ ) y = ๐/๐ โ cot x + ๐ช/(๐ ๐๐๐โก๐ ) Which is the general solution of the given differential equation