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Ex 9.6, 7 - Find general solution x log x dy/dx + y = 2/x log x

Ex 9.6, 7 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.6, 7 - Chapter 9 Class 12 Differential Equations - Part 3 Ex 9.6, 7 - Chapter 9 Class 12 Differential Equations - Part 4 Ex 9.6, 7 - Chapter 9 Class 12 Differential Equations - Part 5


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Ex 9.6, 7 For each of the differential equation given in Exercises 1 to 12, find the general solution : π‘₯π‘™π‘œπ‘”π‘₯ 𝑑𝑦/𝑑π‘₯+𝑦=2/π‘₯ π‘™π‘œπ‘”π‘₯ Step 1: Put in form 𝑑𝑦/𝑑π‘₯ + Py = Q xlog x 𝑑𝑦/𝑑π‘₯ + y = 2/π‘₯ log x Dividing by x log x, 𝑑𝑦/𝑑π‘₯+𝑦" Γ— " 1/(π‘₯ log⁑π‘₯ ) = 2/π‘₯ π‘™π‘œπ‘” π‘₯" Γ— " 1/(π‘₯ log⁑π‘₯ ) 𝑑𝑦/𝑑π‘₯ + (1/(π‘₯ log⁑π‘₯ ))𝑦=2/π‘₯^2 Step 2: Find P and Q ...(1) Comparing (1) with 𝑑𝑦/𝑑π‘₯ + Py = Q P = 1/(π‘₯ log⁑π‘₯ ) & Q = 2/π‘₯2 Step 3: Find Integration factor, I.F IF = e^∫1▒〖𝑝 𝑑π‘₯γ€— IF = e^∫1β–’γ€–1/(π‘₯ log⁑π‘₯ ) 𝑑π‘₯γ€— Let t = log x dt = 1/π‘₯ dx dx = x dt So, IF = e^∫1β–’γ€–1/(π‘₯ 𝑑) Γ— π‘₯𝑑𝑑〗 IF = e^∫1β–’γ€–1/𝑑 𝑑𝑑〗 IF = e^log⁑〖|𝑑|γ€— IF = |𝑑| Putting back t = log x IF = |log x| IF = log x Step 4: Solution of the equation y Γ— I.F = ∫1▒〖𝑄×𝐼.𝐹. 𝑑π‘₯+𝐢〗 Putting values, y Γ— log x = ∫1β–’2/π‘₯2 . log x. dx + C (As log x is always positive) ...(2) Let I = 2 ∫1β–’log⁑〖π‘₯ π‘₯^(βˆ’2) 𝑑π‘₯γ€— Solving I I = 2 ∫1β–’log⁑〖π‘₯ π‘₯^(βˆ’2) 𝑑π‘₯γ€— I = 2["log x. " ∫1β–’γ€–π‘₯^(βˆ’2) 𝑑π‘₯βˆ’βˆ«1β–’ 1/π‘₯ [∫1β–’γ€– π‘₯^(βˆ’2) 𝑑π‘₯γ€—] γ€— 𝑑π‘₯" " ] I = 2 ["log x . " π‘₯^(βˆ’1)/((βˆ’1)) " βˆ’ " ∫1β–’γ€– 1/π‘₯γ€— " . " ((π‘₯^(βˆ’1)))/((βˆ’1)) ".dx " ] = 2["βˆ’ log x. " 1/π‘₯ " + " ∫1β–’γ€–1/π‘₯^2 .𝑑π‘₯γ€—] Integrating by parts with ∫1▒〖𝑓(π‘₯) 𝑔(π‘₯) 𝑑π‘₯=𝑓(π‘₯) ∫1▒〖𝑔(π‘₯) 𝑑π‘₯ βˆ’βˆ«1β–’γ€–[𝑓^β€² (π‘₯) ∫1▒〖𝑔(π‘₯) 𝑑π‘₯] 𝑑π‘₯γ€—γ€—γ€—γ€— Take f (x) = log x & g (x) = π‘₯^(βˆ’2) = 2[(βˆ’1)/π‘₯ " .log x βˆ’ " 1/π‘₯] = (βˆ’2)/π‘₯ (1 + log x) Putting value of I in (2) y log x = I + C y. log x = (βˆ’πŸ)/𝒙 (1 + log x) + C Which is the general solution of the given equation.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.