[SQP] Question 5 - The value of 'n', such that differential equation

Question 5 The value of ' 𝑛 ', such that the differential equation 𝑥^𝑛 𝑑𝑦/𝑑𝑥=𝑦(log 𝑦−log 𝑥+1); (where 𝑥,𝑦∈𝑅^+) is homogeneous, is (A) 0 (B) 1 (C) 2 (D) 3To check homogeneous, we follow these steps Step 1 - Find dy/dx Step 2 - Putting F(x , y) = 𝑑𝑦/𝑑𝑥 and finding F(𝜆x, 𝜆y) If F(𝜆x, 𝜆y) = F(x, y), then it’s a homogenous equation Now, Step 1: Find 𝑑𝑦/𝑑𝑥 𝑥^𝑛 𝑑𝑦/𝑑𝑥=𝑦(log 𝑦−log 𝑥+1) 𝒅𝒚/𝒅𝒙=𝒚/𝒙^𝒏 (𝒍𝒐𝒈 𝒚−𝒍𝒐𝒈 𝒙+𝟏) Step 2: Putting F(x , y) = 𝑑𝑦/𝑑𝑥 and finding F(𝜆x, 𝜆y) F(x, y) = 𝒚/𝒙^𝒏 (𝒍𝒐𝒈 𝒚−𝒍𝒐𝒈 𝒙+𝟏) Finding F(𝝀x, 𝝀y) F(𝜆x, 𝜆y) = 𝝀𝒚/((〖𝝀𝒙)〗^𝒏 )(𝐥𝐨𝐠⁡𝝀𝒚−𝐥𝐨𝐠⁡𝝀𝒙+𝟏) Using log AB = log A + log B = 𝝀/𝝀^𝑛 ×𝒚/𝒙^𝒏 (log⁡𝝀+log⁡𝑦−(log⁡𝝀+log⁡𝑥 )+1) = 𝝀/𝝀^𝑛 ×𝒚/𝒙^𝒏 (log⁡𝝀+log⁡𝑦−log⁡𝝀−log⁡𝑥+1) = 𝝀/𝝀^𝑛 ×𝒚/𝒙^𝒏 (log⁡𝑦−log⁡𝑥+1) Now, F(𝜆x, 𝜆y) = F(x, y) is possible for n = 1 only Thus, for n = 1, the equation is homogeneous So, the correct answer is (B)

Question 5 - CBSE Class 12 Sample Paper for 2025 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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