Ex 9.4, 13 - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
Solving homogeneous differential equation
Ex 9.4, 16 (MCQ)
Ex 9.4, 2
Ex 9.4, 15
Ex 9.4, 4 Important
Example 11
Ex 9.4, 14 Important
Ex 9.4, 13 You are here
Ex 9.4, 8
Ex 9.4, 7 Important
Example 21 Important
Ex 9.4, 6 Important
Ex 9.4, 5
Example 13 Important
Ex 9.4, 9
Ex 9.4, 12 Important
Ex 9.4, 1 Important
Ex 9.4, 3
Ex 9.4, 11
Example 10 Important
Misc 3 Important
Example 12 Important
Ex 9.4, 10 Important
Misc 8 Important
Misc 9
Solving homogeneous differential equation
Last updated at April 16, 2024 by Teachoo
Ex 9.4, 13 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : [๐ฅ sin^2โกใ(๐ฆ/๐ฅ)โ๐ฆใ ]๐๐ฅ+๐ฅ๐๐ฆ=0;๐ฆ=๐/4 When ๐ฅ=1 Ex 9.4, 13 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : [๐ฅ sin^2โกใ(๐ฆ/๐ฅ)โ๐ฆใ ]๐๐ฅ+๐ฅ๐๐ฆ=0;๐ฆ=๐/4 When ๐ฅ=1 Differential equation can be written as [๐ฅ sin^2โกใ(๐ฆ/๐ฅ)โ๐ฆใ ]๐๐ฅ+๐ฅ๐๐ฆ=0 x dy = โ [๐ฅ sin^2โกใ(๐ฆ/๐ฅ)โ๐ฆใ ]dx ๐ ๐/๐ ๐ = โ[๐/๐ ใ๐ฌ๐ข๐งใ^๐โกใ๐/๐ โ๐/๐ใ ] Let F(x, y) = ๐๐ฆ/๐๐ฅ =โ[๐ฅ/๐ฅ sin^2โกใ๐ฆ/๐ฅ โ๐ฆ/๐ฅใ ] Finding F(๐x, ๐y) F(๐x, ๐y) = โ[sin^2โกใ๐๐ฆ/๐๐ฅโ๐๐ฆ/๐๐ฅใ ] = โ[sin^2โกใ๐ฆ/๐ฅโ๐ฆ/๐ฅใ ] = ๐ยฐ F (x, y) โด F(x, y) is ๐ homogenous function of degree zero Putting y = vx Diff w.r.t. x ๐ ๐/๐ ๐ = x ๐ ๐/๐ ๐ + v Putting value of ๐๐ฆ/๐๐ฅ and y = vx in (1) ๐๐ฆ/๐๐ฅ = โ[๐ฅ/๐ฅ sin^2โกใ๐ฆ/๐ฅ โ๐ฆ/๐ฅใ ] v + (๐ ๐ ๐)/๐ ๐ = โ[ใ๐ฌ๐ข๐งใ^๐ ๐๐/๐โ๐๐/๐] v + (๐ฅ ๐๐ฃ)/๐๐ฅ = โ[sin^2 ๐ฃโ๐ฃ] (๐ฅ ๐๐ฃ)/๐๐ฅ = v โ sin2 v โ v (๐ฅ ๐๐ฃ)/๐๐ฅ = โ sin2 v ๐ ๐/ใ๐๐๐ใ^๐โก๐ = โ๐ ๐/๐ Integrating both sides โซ1โ๐๐ฃ/(sin^2 ๐ฃ) = โโซ1โ๐๐ฅ/๐ฅ โซ1โใ๐๐๐ ๐๐^2 ๐ฃ ๐๐ฃใ = โโซ1โ๐๐ฅ/๐ฅ โ cot v = โ log |๐ฅ|+๐ log |๐| โ cot v = C Putting value of v = ๐ฆ/๐ฅ log |๐ฅ| โ cot (๐ฆ/๐ฅ) = C Putting x = 1 & y = ๐ /๐ log 1 โ cot (๐/4) = C 0 โ 1 = C C = โ1 Putting value of C in (2) log |๐ฅ| โ cot ๐ฆ/๐ฅ = โ1 cot ๐ฆ/๐ฅ โ log |๐ฅ| = 1 cot ๐ฆ/๐ฅ โ log |๐ฅ| = log e cot ๐ฆ/๐ฅ = log |๐ฅ| + log e cot ๐/๐ = log |๐๐|