Question 29 (Choice 1) - CBSE Class 12 Sample Paper for 2023 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by Teachoo

Solve the differential equation: ydx+(x-y ² )dy=0

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Question 29 (Choice 1) Solve the differential equation: 𝑦𝑑𝑥+(𝑥−𝑦^2 )𝑑𝑦=0 For equation 𝑦𝑑𝑥+(𝑥−𝑦^2 )𝑑𝑦=0 We observe that we cannot use variable separation method Let’s try to put in the form 𝒅𝒚/𝒅𝒙 + Py = Q or 𝒅𝒙/𝒅𝒚 + P1 x = Q1 Now, y dx + (x − y2) dy = 0 y dx = − (x − y2)dy 𝒅𝒚/𝒅𝒙 = (−𝒚)/(𝒙−𝒚^𝟐 ) This is not of the form 𝑑𝑦/𝑑𝑥 + Py = Q Thus, let’s find 𝒅𝒙/𝒅𝒚 𝑑𝑥/𝑑𝑦 = (𝑦^2 − 𝑥)/𝑦 𝑑𝑥/𝑑𝑦 = y − 𝑥/𝑦 𝒅𝒙/𝒅𝒚 + 𝒙/𝒚 = y Comparing with 𝒅𝒙/𝒅𝒚 + P1 x = Q1 ∴ P1 = 1/𝑦 &. Q1 = y Finding Integrating factor, IF = 𝑒^∫1▒〖𝑝1 𝑑𝑦〗 = 𝑒^∫1▒𝑑𝑦/𝑦 = 𝒆^𝒍𝒐𝒈⁡𝒚 = y Solution is x (IF) = ∫1▒〖(𝑸𝟏 × 𝑰𝑭)𝒅𝒚+𝒄〗 𝑥𝑦=∫1▒〖𝑦 × 𝑦 𝑑𝑦+𝑐〗 𝒙𝒚= ∫1▒〖𝒚^𝟐 𝒅𝒚+𝒄〗 𝒙𝒚= 𝒚^𝟑/𝟑+𝑪

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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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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