Solving homogeneous differential equation

Chapter 9 Class 12 Differential Equations
Concept wise

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Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) (4π₯+6π¦+5)ππ¦β(3π¦+2π₯+4)ππ₯=0 (B) (π₯π¦)ππ₯β(π₯^3+π¦^3 )ππ¦=0 (C) (π₯^3+2π¦^2 )ππ₯+2π₯π¦ ππ¦=0 (D) π¦^2 ππ₯+(π₯^2+π₯π¦βπ¦^2 )ππ¦=0 Let us check each equation one by one Checking (A) Differential equation can be written as (4π₯+6π¦+5)ππ¦β(3π¦+2π₯+4)ππ₯ ππ¦/ππ₯ = ((3π¦ + 2π₯ + 4))/((4π₯ + 6π¦ + 5)) Let F(x, y) = ππ¦/ππ₯ = ((3π¦ + 2π₯ + 4))/((4π₯ + 6π¦ + 5)) Finding F(πx, πy) F(πx, πy) = (2ππ₯ + 3ππ¦ + 4)/(4ππ₯ + 6ππ¦ + 5) β  πΒ° F(x, y) β΄ The given equation is not homogenous Checking (B) (B) Differential equation can be written as (π₯π¦)ππ₯β(π₯^3+π¦^3 )ππ¦ = 0 ππ¦/ππ₯ = π₯π¦/(π₯^3 + π¦^3 ) Let F(x, y) = ππ¦/ππ₯ = π₯π¦/(π₯^3 + π¦^3 ) Finding F(πx, πy) F(πx, πy) = (ππ₯ ππ¦)/(π^3 π₯^3 + π^3 π¦^3 ) = (π^2 π₯π¦)/(π^3 [π₯^3 + π¦^3 ] ) = π₯π¦/π(π₯^3+π¦^3 ) β  πΒ° F(x, y) β΄ The given equation is not homogenous Checking (C) (π₯^3+2π¦^2 )ππ₯+2π₯π¦ ππ¦=0 (x3 + 2y2) dx = β2xy dy ππ¦/ππ₯ = (β(π₯^3 + 2π¦^2))/2π₯π¦ Let F(x, y) = ππ¦/ππ₯ = (β(π₯^3 + 2π¦^2))/2π₯π¦ Finding F(πx, πy) F(πx, πy) = (β(π^3 π₯^3 + 2π^2 π¦^2))/2ππ₯ππ¦ = (βγ6π₯γ^3 + 2π¦^2)/2π₯π¦ β  πΒ° F(x, y) β΄ The given equation is not homogenous Checking (D) y2 dx + (x2 β xy β y2) dy = 0 y2 dx = (x2 β xy β y2)dy ππ¦/ππ₯ = π¦^2/(π₯^2 β π₯π¦ β π¦2) Let F(x, y) = ππ¦/ππ₯ = π¦^2/(π₯^2 β π₯π¦ β π¦2) Finding F(πx, πy) F(πx, πy) = γβπ^(2 ) π¦γ^2/(π^(2 ) (π₯^2 β π₯π¦ β π¦2)) = π¦^2/(π₯^2 β π₯π¦ β π¦2) = πΒ°F (x, y) F (x, y) is π homogenous function of degree zero. β΄ Given equation is a homogenous differential equation. Hence, (D) is the correct answer.