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Example 17 Find the particular solution of the differential equation 𝑑𝑦/𝑑π‘₯+𝑦 cot⁑〖π‘₯=2π‘₯+π‘₯^2 cot⁑π‘₯(π‘₯β‰ 0) γ€— given that 𝑦=0 π‘€β„Žπ‘’π‘› π‘₯=πœ‹/2 𝑑𝑦/𝑑π‘₯+𝑦 cot⁑〖π‘₯=2π‘₯+π‘₯^2 cot⁑π‘₯ γ€— Differential equation is of the form π’…π’š/𝒅𝒙+π‘·π’š=𝑸 where P = cot x & Q = 2x + x2 cot x IF = 𝑒^∫1▒〖𝑝 𝑑π‘₯γ€— IF = 𝒆^∫1β–’γ€–πœπ¨π­β‘π’™ 𝒅𝒙〗 IF = 〖𝑒^log⁑sin⁑π‘₯ γ€—^" " IF = sin x Solution is y (IF) =∫1β–’γ€–(𝑄×𝐼𝐹) 𝑑π‘₯+𝑐〗 y sin x = ∫1▒〖𝐬𝐒𝐧⁑𝒙×(πŸπ’™+𝒙^(𝟐 ) πœπ¨π­β‘π’™ ) 𝒅𝒙〗 + C y sin x = ∫1β–’γ€–(2π‘₯ sin⁑π‘₯+π‘₯^(2 ) sin⁑〖π‘₯ cot⁑π‘₯ γ€— ) 𝑑π‘₯γ€— + C y sin⁑π‘₯ = ∫1β–’γ€–2π‘₯ sin⁑π‘₯ 𝑑π‘₯+γ€— ∫1β–’γ€–π‘₯^2 sin⁑π‘₯ cot⁑π‘₯ 𝑑π‘₯+γ€— 𝐢 y sin⁑π‘₯ = 2∫1▒〖𝐬𝐒𝐧⁑𝒙 (𝒙) 𝒅𝒙〗+∫1β–’γ€–π‘₯^2 sin⁑π‘₯ cot⁑π‘₯ 𝑑π‘₯+γ€— 𝐢 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) = sin x & g (x) = π‘₯ y sin⁑π‘₯ = 2 [𝐬𝐒𝐧⁑𝒙 ∫1▒〖𝒙 π’…π’™βˆ’γ€— ∫1β–’γ€–[𝒄𝒐𝒔⁑〖𝒙 ∫1▒〖𝒙 𝒅𝒙〗 γ€— ] 𝒅𝒙〗] + ∫1β–’γ€–π‘₯^2 sin⁑π‘₯ γ€— cot⁑π‘₯ dx + C y sin⁑π‘₯ = 2 [sin⁑π‘₯ [π‘₯^2/2]βˆ’βˆ«1▒〖𝒄𝒐𝒔⁑〖𝒙 γ€— [π‘₯^2/2]𝒅𝒙〗] + ∫1β–’γ€–π‘₯^2 sin⁑π‘₯ γ€— cot⁑π‘₯ dx y sin x = x2sin x βˆ’ ∫1▒𝒙^𝟐 cos x dx + ∫1▒〖𝒙^𝟐 π’”π’Šπ’β‘π’™ γ€— 𝒄𝒐𝒕⁑𝒙 dx + C y sin x = x2sin x βˆ’ ∫1β–’π‘₯^2 cos x dx + ∫1β–’γ€–π‘₯^2 sin⁑π‘₯ γ€—Γ—cos⁑π‘₯/sin⁑π‘₯ dx + C y sin x = x2sin x βˆ’ ∫1β–’π‘₯^2 cos x dx + ∫1β–’γ€–π‘₯^2 cos⁑π‘₯ γ€— dx + C y sin x = x2 sin x + C Given that y = 0 when x = πœ‹/2 Putting 𝒙=𝝅/𝟐 and y = 0 in (1) (0) sin πœ‹/2=(πœ‹/2)^2 sin⁑〖(πœ‹/2)+Cγ€— 0 =πœ‹^2/4 (1)+C γ€–βˆ’π…γ€—^𝟐/πŸ’=𝐂 Putting value in C in (1) y sin x = x2 sin x + C y sin x = 𝒙^𝟐 π’”π’Šπ’β‘γ€–π’™ βˆ’γ€— 𝝅^𝟐/πŸ’ Dividing both sides by sin x (𝑦 sin⁑π‘₯)/sin⁑π‘₯ =(π‘₯^2 sin⁑π‘₯)/sin⁑π‘₯ βˆ’πœ‹^2/(4 sin⁑π‘₯ ) π’š=𝒙^πŸβˆ’π…^𝟐/γ€–πŸ’ 𝐬𝐒𝐧〗⁑𝒙 where sin⁑〖π‘₯β‰ 0γ€— y sin⁑π‘₯ = 2 [𝐬𝐒𝐧⁑𝒙 ∫1▒〖𝒙 π’…π’™βˆ’γ€— ∫1β–’γ€–[𝒄𝒐𝒔⁑〖𝒙 ∫1▒〖𝒙 𝒅𝒙〗 γ€— ] 𝒅𝒙〗] + ∫1β–’γ€–π‘₯^2 sin⁑π‘₯ γ€— cot⁑π‘₯ dx + C

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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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.