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Misc 13 - Find particular solution: dy/dx + y cot x = 4x cosec x - Solving Linear differential equations - Equation given

Misc 13 - Chapter 9 Class 12 Differential Equations - Part 2
Misc 13 - Chapter 9 Class 12 Differential Equations - Part 3

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Misc 13 Find a particular solution of the differential equation 𝑑𝑦﷮𝑑𝑥﷯+𝑦 cot﷮𝑥=4𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥 𝑥≠0﷯ ,﷯ given that 𝑦=0 when 𝑥= 𝜋﷮2﷯ Given 𝑑𝑦﷮𝑑𝑥﷯+𝑦 cot﷮𝑥=4𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥 ﷯ This of the form 𝑑𝑦﷮𝑑𝑥﷯+𝑃𝑦=𝑄 where P = cot x & Q = 4x cosec x IF = 𝑒﷮ ﷮﷮𝑃𝑑𝑥﷯﷯ IF = 𝑒﷮ ﷮﷮ cot﷮𝑥 𝑑𝑥﷯﷯﷯ IF = 𝑒﷮log⁡( sin﷮𝑥)﷯﷯ IF = sin x Solution is y (IF) = ﷮﷮ 𝑄×𝐼.𝐹﷯𝑑𝑥+𝑐 ﷯ y sin x = ﷮﷮4𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥 sin﷮𝑥﷯ 𝑑𝑥+𝑐 ﷯ y sin x = ﷮﷮4𝑥 1﷮ sin﷮𝑥﷯﷯ sin﷮𝑥﷯𝑑𝑥+𝑐 ﷯ y sin x = ﷮﷮4𝑥 𝑑𝑥+𝑐 ﷯ y sin x = 4 𝑥﷮2﷯﷮2﷯+𝑐 y sin x = 2x2 + C Given that 𝑦=0 when 𝑥= 𝜋﷮2﷯ Put x = 𝜋﷮2﷯ & y = 0 in (1) 0 × sin 𝜋﷮2﷯ = 2 𝜋﷮2﷯﷯﷮2﷯+𝐶 0 = 2 𝜋﷮4﷯﷮2﷯﷯ + C 0 = 𝜋﷮2﷯﷮2﷯ + C C = −𝜋﷮2﷯﷮2﷯ Putting value of C in (1) y sin x = 2x2 + c y sin x = 2 𝒙﷮𝟐﷯ − 𝝅﷮𝟐﷯﷮𝟐﷯

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Davneet Singh

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.