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Ex 9.6, 1 For each of the differential equation given in Exercises 1 to 12, find the general solution : 𝑑𝑦/𝑑π‘₯+2𝑦=𝑠𝑖𝑛π‘₯ Step 1: Put in form 𝑑𝑦/𝑑π‘₯ + Py = Q 𝑑𝑦/𝑑π‘₯+2𝑦=sin⁑π‘₯ Step 2: Find P and Q Comparing (1) with 𝑑𝑦/𝑑π‘₯ + Py = Q ∴ P = 2 and Q = sin x Step 3: Find integrating factor, IF IF = e^∫1▒𝑃𝑑π‘₯ IF = 𝑒^∫1β–’2𝑑π‘₯ IF = 𝒆^πŸπ’™ Step 4 : Solution of the equation y Γ— I.F = ∫1▒〖𝑄×𝐼.𝐹.𝑑π‘₯+𝑐 γ€— Putting values, π’š Γ— 𝒆^πŸπ’™ = ∫1▒〖𝐬𝐒𝐧⁑𝒙 𝒆^πŸπ’™ 𝒅𝒙〗+𝒄 𝐿𝑒𝑑 𝐼= ∫1β–’γ€–sin⁑π‘₯ 𝑒^2π‘₯ 𝑑π‘₯γ€— Solving I 𝐼= ∫1β–’γ€–sin⁑〖π‘₯.𝑒^2π‘₯ γ€—.𝑑π‘₯ γ€— 𝐼 = sin x ∫1▒〖𝒆^πŸπ’™.π’…π’™βˆ’βˆ«1β–’γ€–[(𝒅(π’”π’Šπ’β‘π’™))/𝒅𝒙 ∫1▒〖𝒆^πŸπ’™ 𝒅𝒙 γ€—] γ€— γ€— 𝐼 = sin x 𝑒^2π‘₯/2 βˆ’ ∫1β–’cos⁑π‘₯ 𝑒^2π‘₯/2 dx 𝐼 = 𝟏/𝟐 π’”π’Šπ’β‘γ€–π’™ 𝒆^πŸπ’™ γ€—βˆ’πŸ/𝟐 [𝒄𝒐𝒔⁑𝒙 ∫1▒𝒆^πŸπ’™ 𝒅𝒙 βˆ’βˆ«1β–’(𝒅(𝒄𝒐𝒔⁑𝒙))/𝒅𝒙 ∫1▒𝒆^πŸπ’™ 𝒅𝒙 ]dx 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) = 𝑒^2π‘₯ Again using 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^2π‘₯ 𝐼 = 1/2 sin⁑〖π‘₯ 𝑒^2π‘₯ γ€—βˆ’1/2 [cos⁑π‘₯ ∫1▒𝑒^2π‘₯/2 βˆ’βˆ«1β–’γ€–(βˆ’sin x)γ€— ∫1▒𝑒^2π‘₯/2 𝑑π‘₯ ] 𝐼 = 1/2 sin⁑〖π‘₯ 𝑒^2π‘₯ γ€—βˆ’1/2 [cos⁑π‘₯ 𝑒^2π‘₯/2+1/2 ∫1▒〖𝐬𝐒𝐧 𝐱 𝒆^πŸπ’™ 𝒅𝒙〗] 𝐼 = 1/2 sin⁑〖π‘₯ 𝑒^2π‘₯ γ€—βˆ’1/2 [(cos⁑π‘₯ 𝑒^2π‘₯)/2+1/2 𝑰] + C I = 1/2 sin x 𝑒^2π‘₯ βˆ’1/4 cos x 𝑒^2π‘₯ βˆ’ 𝟏/πŸ’ I + C I + 𝟏/πŸ’ I = 1/4 [2 sin⁑〖π‘₯ 𝑒^2π‘₯ βˆ’cos⁑〖π‘₯ 𝑒^2π‘₯ γ€— γ€— ] + C 5𝐼/4 = 𝑒^2π‘₯/4 [2 sin⁑〖π‘₯βˆ’cos⁑π‘₯ γ€— ] + C 𝑰 = 𝒆^πŸπ’™/πŸ“ [𝟐 π’”π’Šπ’β‘γ€–π’™βˆ’π’„π’π’”β‘π’™ γ€— ] + C Now, Putting value of I in (2) y 𝑒^2π‘₯ = 𝑒^2π‘₯/5 [2 sin⁑〖π‘₯ βˆ’cos⁑π‘₯ γ€— ] + C Dividing by e2x y = 𝟏/πŸ“ [𝟐 𝐬𝐒𝐧⁑〖𝒙 βˆ’πœπ¨π¬β‘π’™ γ€— ]+π‘ͺ𝒆^(βˆ’πŸπ’™)

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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.