Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
Ex 9.5
Ex 9.5, 2
Ex 9.5, 3 Important
Ex 9.5, 4
Ex 9.5, 5 Important
Ex 9.5, 6
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
Ex 9.5, 13
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 April 16, 2024 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