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Ex 9.5

Ex 9.5, 1
Important

Ex 9.5, 2

Ex 9.5, 3 Important

Ex 9.5, 4

Ex 9.5, 5 Important

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Ex 9.5, 7 Important

Ex 9.5, 8 Important

Ex 9.5, 9

Ex 9.5, 10 You are here

Ex 9.5, 11

Ex 9.5, 12 Important

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Ex 9.5, 14 Important

Ex 9.5, 15

Ex 9.5, 16 Important

Ex 9.5, 17 Important

Ex 9.5, 18 (MCQ)

Ex 9.5, 19 (MCQ) Important

Last updated at Aug. 14, 2023 by Teachoo

Ex 9.5, 10 For each of the differential equation given in Exercises 1 to 12, find the general solution : (𝑥+𝑦) 𝑑𝑦/𝑑𝑥=1 Step 1: Put in form 𝑑𝑦/𝑑𝑥 + Py = Q or 𝑑𝑥/𝑑𝑦 + P1x = Q1 (x + y) 𝑑𝑦/𝑑𝑥 = 1 Dividing by (x + y), 𝑑𝑦/𝑑𝑥 = 1/((𝑥 + 𝑦)) 𝑑𝑥/𝑑𝑦 = (𝑥+𝑦) 𝑑𝑥/𝑑𝑦 − x = 𝑦 𝒅𝒙/𝒅𝒚 + (−1) x = 𝒚 Step 2: Find P1 and Q1 Comparing (1) with 𝑑𝑥/𝑑𝑦 + P1x = Q1 P1 = −1 and Q1 = y Step 3: Find Integrating factor, I.F. I.F. = 𝑒^∫1▒〖𝑃_1 𝑑𝑦〗 = e^∫1▒〖(−1) 𝑑𝑦〗 = e−y Step 4 : Solution of the equation x × I.F. = ∫1▒〖𝑄1×𝐼.𝐹. 𝑑𝑦+𝐶〗 Putting values, x × e − y = ∫1▒〖𝑦 × 𝑒^(−𝑦).〗 𝑑𝑦+𝐶 Let I = ∫1▒〖𝒚.𝒆^(−𝒚) 𝒅𝒚〗 Using Integration by parts ∫1▒〖𝑓(𝑦)𝑔(𝑦)𝑑𝑦=𝑓(𝑦) ∫1▒〖𝑔(𝑦)𝑑𝑦−〗〗 ∫1▒[𝑓′(𝑦)∫1▒𝑔(𝑦)𝑑𝑦] 𝑑𝑦 Taking 𝑓(𝑦) = y and g (y) = 𝑒^(−𝑦) = y ∫1▒〖𝒆^(−𝒚) 𝒅𝒚〗 − ∫1▒[𝒅/𝒅𝒚 𝒚 ∫1▒〖𝒆^(−𝒚) 𝒅𝒚〗] dy = y 𝑒^(−𝑦)/(−1) − ∫1▒〖1. 〗 𝑒^(−𝑦)/(−1) dy. = −𝑦.𝑒^(−𝑦) + ∫1▒〖𝑒^(−𝑦) 𝑑𝑦〗 = −𝑦.𝑒^(−𝑦) + 𝑒^(−𝑦)/(−1) = −𝑦.𝑒^(−𝑦) – 𝑒^(−𝑦) Putting value of I in (2) x e −y = ∫1▒〖𝑦×𝑒^(−𝑦).〗 𝑑𝑦+𝐶 x e −y = −𝒚𝒆^(−𝒚)−𝒆^(−𝒚)+𝑪 Dividing by 𝑒^(−𝑦) x = −y − 1 + Cey x + y + 1 = Cey