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Transcript

Example 15 Find the general solution of the differential equation π‘₯ 𝑑𝑦/𝑑π‘₯+2𝑦=π‘₯^2 (π‘₯β‰ 0) π‘₯ 𝑑𝑦/𝑑π‘₯+2𝑦=π‘₯^2 (π‘₯ 𝑑𝑦)/(π‘₯ 𝑑π‘₯) + 2𝑦/π‘₯ = π‘₯^2/π‘₯ Dividing both sides by x π’…π’š/𝒅𝒙 + πŸπ’š/𝒙 = x Differential equation is of the form 𝑑𝑦/𝑑π‘₯+𝑃𝑦=𝑄 where P = 2/π‘₯ & Q = x Finding Integrating Factor IF = 𝑒^∫1▒〖𝑝 𝑑π‘₯γ€— IF = 𝒆^∫1β–’γ€–πŸ/𝒙 𝒅𝒙〗 IF = 𝑒^(2 log⁑π‘₯ ) IF = 𝑒^(γ€–log⁑π‘₯γ€—^2 ) I.F = x2 Solution of differential equation is y Γ— IF = ∫1β–’γ€–(π‘ΈΓ—πˆπ…)𝒅𝒙+𝒄〗 yx2 = ∫1β–’γ€–π‘₯Γ—π‘₯^2 𝑑π‘₯+𝑐〗 yx2 = ∫1▒〖𝒙^πŸ‘ 𝒅𝒙+𝒄〗 x2 y = π‘₯^4/4+𝑐 y = 𝒙^𝟐/πŸ’+𝒄𝒙^(βˆ’πŸ) is the required general solution

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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, Social Science, Physics, Chemistry, Computer Science at Teachoo.