Ex 9.5, 9 - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
Ex 9.5
Ex 9.5, 2
Ex 9.5, 3 Important
Ex 9.5, 4
Ex 9.5, 5 Important
Ex 9.5, 6
Ex 9.5, 7 Important
Ex 9.5, 8 Important
Ex 9.5, 9 You are here
Ex 9.5, 10
Ex 9.5, 11
Ex 9.5, 12 Important
Ex 9.5, 13
Ex 9.5, 14 Important
Ex 9.5, 15
Ex 9.5, 16 Important
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