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Ex 9.4, 15 - Find equation of curve passing through (0, 0)

Ex 9.4, 15 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.4, 15 - Chapter 9 Class 12 Differential Equations - Part 3
Ex 9.4, 15 - Chapter 9 Class 12 Differential Equations - Part 4

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Ex 9.4, 15 Find the equation of curve passing through the point (0 , 0) and whose differential equation is 𝑦^β€²=𝑒^π‘₯ sin⁑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^π‘₯ sin x 𝑑𝑦 = 𝑒^π‘₯ sin x dx Integrating both sides ∫1▒𝑑𝑦 = ∫1▒〖𝑒π‘₯ sin⁑〖π‘₯ 𝑑π‘₯γ€— γ€— y = ∫1▒〖𝑒π‘₯ sin⁑〖π‘₯ 𝑑π‘₯γ€— γ€— …(1) Integrating by parts, using formula ∫1▒〖𝑓 (π‘₯)𝑔(π‘₯)𝑑π‘₯ γ€—= f(π‘₯) ∫1▒〖𝑔(π‘₯)𝑑π‘₯ βˆ’βˆ«1β–’γ€–[𝑓′(π‘₯)∫1▒𝑔(π‘₯)𝑑π‘₯] 𝑑π‘₯ γ€— γ€— Taking f(x) = ex and g (x) = sin x y = 𝑒^π‘₯ ∫1▒〖〖𝑠𝑖𝑛 π‘₯〗⁑〖 𝑑π‘₯γ€—βˆ’γ€— ∫1β–’[(𝑒^π‘₯) ∫1β–’γ€–sin⁑π‘₯ 𝑑π‘₯ γ€—]𝑑π‘₯ y = 𝑒^π‘₯ (βˆ’cos⁑π‘₯) βˆ’ ∫1β–’γ€–(βˆ’cos⁑π‘₯)γ€— 𝑒^π‘₯ 𝑑π‘₯ y = –𝑒^π‘₯ π‘π‘œπ‘ β‘π‘₯+∫1▒〖𝑒^π‘₯ cos⁑π‘₯ 𝑑π‘₯γ€— Integrating by parts, using formula ∫1▒〖𝑓 (π‘₯)𝑔(π‘₯)𝑑π‘₯ γ€—= f(π‘₯) ∫1▒〖𝑔(π‘₯)𝑑π‘₯ βˆ’βˆ«1β–’γ€–[𝑓′(π‘₯)∫1▒𝑔(π‘₯)𝑑π‘₯] 𝑑π‘₯ γ€— γ€— Taking f(x) = ex and g(x) = cos x y = –𝑒^π‘₯ π‘π‘œπ‘ β‘π‘₯+ 𝑒^π‘₯ ∫1β–’γ€–γ€–π‘π‘œπ‘  π‘₯〗⁑〖 𝑑π‘₯γ€—βˆ’γ€— ∫1β–’[(𝑒^π‘₯) ∫1β–’γ€–cos⁑π‘₯ 𝑑π‘₯ γ€—]𝑑π‘₯ y = –𝑒^π‘₯ π‘π‘œπ‘ β‘π‘₯+𝑒^π‘₯ 𝑠𝑖𝑛⁑π‘₯ βˆ’βˆ«1▒〖𝑒^π‘₯ sin⁑π‘₯ 𝑑π‘₯γ€— But From (1) y = ∫1▒〖𝑒π‘₯ 𝑠𝑖𝑛⁑〖π‘₯ 𝑑π‘₯γ€— γ€— y = –𝑒^π‘₯ π‘π‘œπ‘ β‘π‘₯+𝑒^π‘₯ 𝑠𝑖𝑛⁑π‘₯ βˆ’π‘¦ y + y = –𝑒^π‘₯ π‘π‘œπ‘ β‘π‘₯+𝑒^π‘₯ 𝑠𝑖𝑛⁑π‘₯ 2y = 𝑒^π‘₯ (sin⁑π‘₯βˆ’cos⁑π‘₯) y = 1/2 𝑒^π‘₯ (sin⁑π‘₯βˆ’cos⁑π‘₯ ) + C Given curve passes through (0, 0) Putting x = 0, y = 0 in equation 0 = 1/2 𝑒^0 (sin⁑0βˆ’cos⁑0) + C 0 = 1/2 (0βˆ’1) + C 0 = (βˆ’1)/2 + C C = 1/2 Putting value of C in (1) y = 1/2 𝑒^π‘₯ (sin⁑π‘₯βˆ’cos⁑π‘₯ ) + 1/2 y – 1/2= 1/2 𝑒^π‘₯ (sin⁑π‘₯βˆ’cos⁑π‘₯ ) (2𝑦 βˆ’ 1)/2= 1/2(𝑒^π‘₯ sin⁑π‘₯βˆ’π‘’^π‘₯ cos⁑π‘₯) 2y – 1 = (𝒆^𝒙 π’”π’Šπ’β‘π’™βˆ’π’†^𝒙 𝒄𝒐𝒔⁑𝒙)

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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.