

CBSE Class 12 Sample Paper for 2025 Boards
CBSE Class 12 Sample Paper for 2025 Boards
Last updated at Dec. 13, 2024 by Teachoo
Question 5 The value of ' π ', such that the differential equation π₯^π ππ¦/ππ₯=π¦(log π¦βlog π₯+1); (where π₯,π¦βπ ^+) 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) = ππ¦/ππ₯ and finding F(πx, πy) If F(πx, πy) = F(x, y), then itβs a homogenous equation Now, Step 1: Find ππ¦/ππ₯ π₯^π ππ¦/ππ₯=π¦(log π¦βlog π₯+1) π π/π π=π/π^π (πππ πβπππ π+π) Step 2: Putting F(x , y) = ππ¦/ππ₯ and finding F(πx, πy) F(x, y) = π/π^π (πππ πβπππ π+π) Finding F(πx, πy) F(πx, πy) = ππ/((γππ)γ^π )(π₯π¨π β‘ππβπ₯π¨π β‘ππ+π) Using log AB = log A + log B = π/π^π Γπ/π^π (logβ‘π+logβ‘π¦β(logβ‘π+logβ‘π₯ )+1) = π/π^π Γπ/π^π (logβ‘π+logβ‘π¦βlogβ‘πβlogβ‘π₯+1) = π/π^π Γπ/π^π (logβ‘π¦βlogβ‘π₯+1) Now, F(πx, πy) = F(x, y) is possible for n = 1 only Thus, for n = 1, the equation is homogeneous So, the correct answer is (B)