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Last updated at May 29, 2018 by Teachoo
Transcript
Ex 9.6, 11 For each of the differential equation find the general solution : 𝑦 𝑑𝑥+ 𝑥− 𝑦2𝑑𝑦=0 Step 1 : Put in form 𝑑𝑦𝑑𝑥 + Py = Q or 𝑑𝑥𝑑𝑦 + P1 x = Q1, y dx + (x − y2) dy = 0 y dx = − (x − y2)dy 𝑑𝑦𝑑𝑥 = −𝑦𝑥− 𝑦2 This is not of the form 𝑑𝑦𝑑𝑥 + Py = Q ∴ we find 𝑑𝑥𝑑𝑦 𝑑𝑥𝑑𝑦 = 𝑦2 − 𝑥𝑦 𝑑𝑥𝑑𝑦 = y − 𝑥𝑦 𝑑𝑥𝑑𝑦 + 𝑥𝑦 = y Step 2 : Find P1 and Q1 Comparing (1) with 𝑑𝑥𝑑𝑦 + P1 x = Q1 Where P1 = 1𝑦 & Q1 = y Step 3 : Find Integrating factor, IF = 𝑒 𝑝1 𝑑𝑦 = 𝑒 𝑑𝑦𝑦 = 𝑒 log𝑦 = y Step 4 : Solution is x (IF) = 𝑄1×𝐼𝐹𝑑𝑦+𝑐 xy = 𝑦×𝑦 𝑑𝑦+𝑐 xy = 𝑦2 𝑑𝑦+𝑐 xy = 𝑦33+𝐶 x = 𝑦33𝑦+ 𝐶𝑦 x = 𝒚𝟐𝟑+ 𝑪𝒚
Ex 9.6
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