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Solving homogeneous differential equation
Ex 9.5, 16 (MCQ)
Ex 9.5, 2
Ex 9.5, 15
Ex 9.5, 4 Important
Example 16
Ex 9.5, 14 Important
Ex 9.5, 13
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Ex 9.5, 7 Important
Example 27 Important
Ex 9.5, 6 Important
Ex 9.5, 5
Example 18 Important
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Ex 9.5, 12 Important
Ex 9.5, 1 Important
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Ex 9.5, 11
Example 15 Important
Misc 4 Important
Example 17 Important
Ex 9.5, 10 Important
Misc 10 Important
Misc 11
Solving homogeneous differential equation
Last updated at Aug. 11, 2021 by Teachoo
Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) (4π₯+6π¦+5)ππ¦β(3π¦+2π₯+4)ππ₯=0 (B) (π₯π¦)ππ₯β(π₯^3+π¦^3 )ππ¦=0 (C) (π₯^3+2π¦^2 )ππ₯+2π₯π¦ ππ¦=0 (D) π¦^2 ππ₯+(π₯^2+π₯π¦βπ¦^2 )ππ¦=0 Let us check each equation one by one Checking (A) Differential equation can be written as (4π₯+6π¦+5)ππ¦β(3π¦+2π₯+4)ππ₯ ππ¦/ππ₯ = ((3π¦ + 2π₯ + 4))/((4π₯ + 6π¦ + 5)) Let F(x, y) = ππ¦/ππ₯ = ((3π¦ + 2π₯ + 4))/((4π₯ + 6π¦ + 5)) Finding F(πx, πy) F(πx, πy) = (2ππ₯ + 3ππ¦ + 4)/(4ππ₯ + 6ππ¦ + 5) β πΒ° F(x, y) β΄ The given equation is not homogenous Checking (B) (B) Differential equation can be written as (π₯π¦)ππ₯β(π₯^3+π¦^3 )ππ¦ = 0 ππ¦/ππ₯ = π₯π¦/(π₯^3 + π¦^3 ) Let F(x, y) = ππ¦/ππ₯ = π₯π¦/(π₯^3 + π¦^3 ) Finding F(πx, πy) F(πx, πy) = (ππ₯ ππ¦)/(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 πΒ° F(x, y) β΄ The given equation is not homogenous Checking (C) (π₯^3+2π¦^2 )ππ₯+2π₯π¦ ππ¦=0 (x3 + 2y2) dx = β2xy dy ππ¦/ππ₯ = (β(π₯^3 + 2π¦^2))/2π₯π¦ Let F(x, y) = ππ¦/ππ₯ = (β(π₯^3 + 2π¦^2))/2π₯π¦ Finding F(πx, πy) F(πx, πy) = (β(π^3 π₯^3 + 2π^2 π¦^2))/2ππ₯ππ¦ = (βγ6π₯γ^3 + 2π¦^2)/2π₯π¦ β πΒ° 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 ππ¦/ππ₯ = π¦^2/(π₯^2 β π₯π¦ β π¦2) Let F(x, y) = ππ¦/ππ₯ = π¦^2/(π₯^2 β π₯π¦ β π¦2) Finding F(πx, πy) F(πx, πy) = γβπ^(2 ) π¦γ^2/(π^(2 ) (π₯^2 β π₯π¦ β π¦2)) = π¦^2/(π₯^2 β π₯π¦ β π¦2) = πΒ°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.