Ex 9.3, 7 - Chapter 9 Class 12 Differential Equations

Last updated at Dec. 16, 2024 by Teachoo

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Ex 9.3, 7 For each of the differential equations in Exercises 1 to 10, find the general solution : 𝑦 log⁡〖𝑦 𝑑𝑥 −𝑥 𝑑𝑦=0〗 𝑦 log⁡〖𝑦 𝑑𝑥 −𝑥 𝑑𝑦=0〗 𝑦 log⁡𝑦 𝑑𝑥=𝑥 𝑑𝑦 𝒅𝒙/𝒙 = 𝒅𝒚/(𝒚 𝐥𝐨𝐠⁡𝒚 ) Integrating both sides ∫1▒〖𝑑𝑦/(𝑦 log⁡𝑦 )= ∫1▒𝑑𝑥/𝑥〗 ∫1▒𝒅𝒚/(𝒚 𝒍𝒐𝒈⁡𝒚 )=𝐥𝐨𝐠⁡〖|𝒙|〗+𝑪 Putting t = log y dt = 1/𝑦 dy dy = y dt Hence, our equation becomes ∫1▒(𝑦 𝑑𝑡)/(𝑦.𝑡)=log⁡〖|𝑥|〗+𝐶 ∫1▒𝑑𝑡/𝑡=log⁡〖|𝑥|〗+𝐶 𝑙𝑜𝑔 |𝑡|=𝑙𝑜𝑔⁡〖|𝑥|〗+𝐶 𝒍𝒐𝒈 |𝒕|=𝒍𝒐𝒈⁡|𝒙|+𝐥𝐨𝐠⁡𝑪 Putting t = log y log (log y) = log x + log c log (log y) = log cx (Using log ab = log 𝑎 + log b) Cancelling log log y = cx y = ecx

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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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