Example 19 - Find general solution: dy/dx - y = cos x - Examples

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise

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Example 19 Find the general solution of the differential equation 𝑑𝑦/𝑑π‘₯βˆ’π‘¦=cos⁑π‘₯ Differential equation is of the form 𝑑𝑦/𝑑π‘₯+𝑃𝑦=𝑄 where P = βˆ’1 & Q = cos x IF = e^∫1▒𝑝𝑑π‘₯ IF = e^(βˆ’βˆ«1β–’1𝑑π‘₯) IF = 𝑒^(βˆ’π‘₯) Solution is y(IF) = ∫1β–’γ€–(𝑄×𝐼𝐹) 𝑑π‘₯+𝑐〗 𝑦𝑒^(βˆ’π‘₯) = ∫1▒𝑒^(βˆ’π‘₯) cos⁑〖π‘₯+𝑐〗 Let I = ∫1▒𝑒^(βˆ’π‘₯) cos⁑〖π‘₯ 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) = cos x & g (x) = 𝑒^"βˆ’x" I = cos x ∫1▒〖𝑒^(βˆ’π‘₯) 𝑑π‘₯γ€—βˆ’ ∫1β–’[βˆ’sin⁑〖π‘₯∫1▒〖𝑒^(βˆ’π‘₯) 𝑑π‘₯γ€—γ€— ]𝑑π‘₯ I = γ€–βˆ’π‘’γ€—^(βˆ’π‘₯)cos x βˆ’βˆ«1β–’γ€–βˆ’sin⁑〖π‘₯ (βˆ’π‘’^(βˆ’π‘₯) γ€—)γ€— 𝑑π‘₯ I = βˆ’eβˆ’x cos x βˆ’ ∫1▒〖𝑒^(βˆ’π‘₯) sin⁑〖π‘₯ 𝑑π‘₯γ€— 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) = sin x g (x) = eβˆ’x I = βˆ’eβˆ’x cos x βˆ’ [sin⁑〖π‘₯ ∫1▒〖𝑒^(βˆ’π‘₯) 𝑑π‘₯γ€—βˆ’βˆ«1β–’γ€–(cos〗⁑〖π‘₯ ∫1▒〖𝑒^(βˆ’π‘₯) 𝑑π‘₯γ€—)γ€— "dx " γ€— ] I = βˆ’eβˆ’x cos x βˆ’ [βˆ’π‘’^(βˆ’π‘₯) sin⁑〖π‘₯ βˆ’βˆ«1β–’γ€–βˆ’π‘’^(βˆ’π‘₯) cos⁑π‘₯ 𝑑π‘₯γ€— " " γ€— ] I = βˆ’eβˆ’x cos x βˆ’ [βˆ’π‘’^(βˆ’π‘₯) sin⁑〖π‘₯+∫1▒〖𝑒^(βˆ’π‘₯) cos⁑π‘₯ 𝑑π‘₯γ€— " " γ€— ] I = βˆ’eβˆ’x cos x + 𝑒^(βˆ’π‘₯) sin⁑〖π‘₯βˆ’βˆ«1▒〖𝒆^(βˆ’π’™) 𝒄𝒐𝒔⁑𝒙 𝒅𝒙〗 " " γ€— I = eβˆ’x (sin x βˆ’ cos x) βˆ’ I 2I = eβˆ’x (sin x βˆ’ cos x) I = 𝑒^(βˆ’π‘₯)/2 (sin x βˆ’ cos x) From (1) y 𝑒^(βˆ’π‘₯) = ∫1▒〖𝑒^(βˆ’π‘₯) cos⁑〖π‘₯+𝑐〗 γ€— y 𝑒^(βˆ’π‘₯) = 𝑒^(βˆ’π‘₯)/2 (sin x βˆ’ cos x) + c y = 𝟏/𝟐 (sin x βˆ’ cos x) + c𝒆^𝒙

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.