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Ex 9.6, 15 - Find particular solution: dy/dx - 3y cot x = sin 2x

Ex 9.6, 15 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.6, 15 - Chapter 9 Class 12 Differential Equations - Part 3 Ex 9.6, 15 - Chapter 9 Class 12 Differential Equations - Part 4 Ex 9.6, 15 - Chapter 9 Class 12 Differential Equations - Part 5


Transcript

Ex 9.6, 15 For each of the differential equations given in Exercises 13 to 15 , find a particular solution satisfy the given condition : 𝑑𝑦/𝑑π‘₯βˆ’3𝑦 cot⁑〖π‘₯=sin⁑〖2π‘₯;𝑦=2γ€— γ€— when π‘₯= πœ‹/2 𝑑𝑦/𝑑π‘₯βˆ’3𝑦 cot⁑〖π‘₯=sin⁑2π‘₯ γ€— 𝑑𝑦/𝑑π‘₯ + (βˆ’3 cot x) y = (sin 2x) Comparing with 𝑑𝑦/𝑑π‘₯ + Py = Q P = βˆ’3 cot x & Q = sin 2x Finding Integrating factor (IF) IF = e^∫1▒𝑝𝑑π‘₯ …(1) = e^∫1β–’γ€–βˆ’3 cot⁑〖π‘₯ 𝑑π‘₯γ€— γ€— = e^(βˆ’3∫1β–’γ€–π‘π‘œπ‘‘ π‘₯ 𝑑π‘₯γ€—) = e^(βˆ’3 log⁑|sin⁑π‘₯ | ) = e^log⁑〖|sin⁑π‘₯ |^(βˆ’3) γ€— = e^log⁑〖1/|sin^3⁑π‘₯ | γ€— = e^log⁑|π‘π‘œπ‘ π‘’π‘^3 π‘₯| = π‘π‘œπ‘ π‘’π‘^3 π‘₯ ∴ I.F = π‘π‘œπ‘ π‘’π‘^3 π‘₯ Solution of differential equation is y Γ— IF = ∫1▒〖𝑄.𝐼𝐹 𝑑π‘₯γ€— Putting values y Γ— cosec3 x = ∫1β–’sin⁑〖2π‘₯. π‘π‘œπ‘ π‘’π‘^3 π‘₯ 𝑑π‘₯γ€— y cosec3x = ∫1β–’(2 sin⁑〖π‘₯ cos⁑π‘₯ γ€—)/sin^3⁑π‘₯ dx y cosec3x = ∫1β–’(2 cos⁑π‘₯)/sin^2⁑π‘₯ dx y cosec3x = 2∫1β–’γ€–cos⁑π‘₯/𝑠𝑖𝑛⁑π‘₯ Γ—1/sin⁑π‘₯ γ€— dx y cosec3x = 2∫1β–’γ€–cot⁑π‘₯ π‘π‘œπ‘ π‘’π‘ π‘₯γ€— dx y cosec3x = 2 (βˆ’cosec x) + C y = (βˆ’2 π‘π‘œπ‘ π‘’π‘ π‘₯)/(π‘π‘œπ‘ π‘’π‘^3 π‘₯) + 𝐢/(π‘π‘œπ‘ π‘’π‘^3 π‘₯) y = (βˆ’2)/(π‘π‘œπ‘ π‘’π‘^2 π‘₯) + 𝐢/(π‘π‘œπ‘ π‘’π‘^3 π‘₯) y = βˆ’2 sin2 x + C sin3 x Putting x = πœ‹/2 , y = 2 in (2) 2 = βˆ’2 sin2 πœ‹/2 + C sin3 πœ‹/2 2 = βˆ’2 (1)2 + C(1)3 2 = βˆ’2 + C C = 2 + 2 C = 4 …(2) Put value of C in (3) y = βˆ’2 sin2 x + C sin3 x y = βˆ’2 sin2 x + 4 sin3 x y = 4 sin3 x βˆ’ 2 sin2 x

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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.