Ex 9.3, 7 - Chapter 9 Class 12 Differential Equations

Last updated at April 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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