# Ex 9.6, 15

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 9.6, 15For each of the differential equations given in Exercises 13 to 15 , find a particular solution satisfy the given condition :ππ¦/ππ₯β3π¦ cotβ‘γπ₯=sinβ‘γ2π₯;π¦=2γ γ when π₯= π/2 ππ¦/ππ₯β3π¦ cotβ‘γπ₯=sinβ‘2π₯ γ ππ¦/ππ₯ + (β3 cot x) y = (sin 2x) Comparing with ππ¦/ππ₯ + Py = Q P = β3 cot x & Q = sin 2x Finding Integrating factor (IF) IF = e^β«1βπππ₯ = e^β«1βγβ3 cotβ‘γπ₯ ππ₯γ γ = e^(β3β«1βγπππ‘ π₯ ππ₯γ) = e^(3 logβ‘|sinβ‘π₯ | ) = e^logβ‘γ|sinβ‘π₯ |^(β3) γ = e^logβ‘γ1/|sin^3β‘π₯ | γ = e^logβ‘γ|πππ ππ^3 π₯|^3 γ = πππ ππ^3 π₯ β΄ I.F = πππ ππ^3 π₯ Solution of differential equation y Γ IF = β«1βγπ.πΌπΉ ππ₯γ Putting values. y Γ cosec3 x = β«1βsinβ‘γ2π₯. πππ ππ^3 π₯ ππ₯γ y cosec3x = β«1β(2 sinβ‘γπ₯ cosβ‘π₯ γ)/sin^3β‘π₯ dx y cosec3x = β«1β(2 cosβ‘π₯)/sin^2β‘π₯ dx Let I = 2β«1βcosβ‘π₯/sin^2β‘π₯ ππ₯ Put t = sin x Diff w.r.t x ππ‘/ππ₯ = cos x dx = ππ‘/cosβ‘π₯ β΄ I = 2β«1βcosβ‘π₯/(π‘^2 ) ππ‘/(πππ π₯) I = 2β«1βππ‘/(π‘^2 ) = 2 γπ‘/(β1 )γ^(β1)+π =β2/π‘+π Put value t = sin x = (β2)/sinβ‘π₯ +π =β2 πππ ππ π₯+π Put value of I in (2) β΄ y cosec3 x = β2 cosec x + c y = (β2 πππ ππ π₯)/(πππ ππ^2 π₯) + π/(πππ ππ^3 π₯) y = 2 sin2 x + C sin3 x Putting x = π/2 , y = 2 in (3) 2 = β2 Sin2 π/2 + C Sin3 π/2 2 = β2 (1)2 + C(1)3 2 = β2 + C C = 2 + 2 C = 4 Put value of C in (3) y = β2 sin2 x + C sin3 x y = β2 sin2 x + 4 sin3 x y = 4 sin3 x β 2 sin2 x

Chapter 9 Class 12 Differential Equations

Serial order wise

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.