





Last updated at Dec. 11, 2019 by Teachoo
Ex 9.5, 10 In each of the Exercise 1 to 10 , show that the given differential equation is homogeneous and solve each of them. (1+๐^(๐ฅ/๐ฆ) )๐๐ฅ+๐^(๐ฅ/๐ฆ) (1โ๐ฅ/๐ฆ)๐๐ฆ=0 Since the equation is in the form ๐ฅ/๐ฆ , we will take ๐๐ฅ/๐๐ฆ Instead of ๐๐ฆ/๐๐ฅ Step 1 : Find ๐๐ฅ/๐๐ฆ (1+๐^(๐ฅ/๐ฆ) )๐๐ฅ+๐^(๐ฅ/๐ฆ) (1โ๐ฅ/๐ฆ)๐๐ฆ = 0 (1+๐^(๐ฅ/๐ฆ) ) dx = โ๐^(๐ฅ/๐ฆ) (1โ๐ฅ/๐ฆ)๐๐ฆ ๐๐ฅ/๐๐ฆ = (โ๐^(๐ฅ/๐ฆ) (1 โ ๐ฅ/๐ฆ) )/(1 + ๐^(๐ฅ/๐ฆ) ) โฆ(1) Step 2: Put ๐๐ฅ/๐๐ฆ = F(x, y) and find F(๐x, ๐y) F(x, y) = (โ๐^(๐ฅ/๐ฆ) (1 โ ๐ฅ/๐ฆ) )/(1 + ๐^(๐ฅ/๐ฆ) ) F(๐x, ๐y) = (โ๐^(๐๐ฅ/๐๐ฆ) (1 โ ๐๐ฅ/๐๐ฆ) )/(1 + ๐^(๐๐ฅ/๐๐ฆ) ) = (โ๐^(๐ฅ/๐ฆ) (1 โ ๐ฅ/๐ฆ) )/(1 + ๐^(๐ฅ/๐ฆ) ) = ๐น(๐ฅ, ๐ฆ) So, F(๐๐ฅ ,๐๐ฆ)= F(๐ฅ , ๐ฆ) = ๐ยฐ F(๐ฅ , ๐ฆ) Thus , F(๐ฅ ,๐ฆ) is a homogeneous function of degree zero Therefore given differential equation is homogeneous differential equation Step 3: Solving ๐๐ฅ/๐๐ฆ by Putting ๐ฅ=๐ฃ๐ฆ Putting ๐ฅ=๐ฃ๐ฆ Diff. w.r.t. ๐ฆ ๐๐ฅ/๐๐ฆ=๐/๐๐ฆ (๐ฃ๐ฆ) ๐๐ฅ/๐๐ฆ=๐ฆ . ๐๐ฃ/๐๐ฆ+๐ฃ ๐๐ฆ/๐๐ฆ ๐๐ฅ/๐๐ฆ=๐ฆ . ๐๐ฃ/๐๐ฆ+๐ฃ Putting values of ๐๐ฅ/๐๐ฆ and x in (1) ๐๐ฅ/๐๐ฆ=(โ๐^(๐ฅ/๐ฆ) (1 โ ๐ฅ/๐ฆ) )/(1 + ๐^(๐ฅ/๐ฆ) ) ๐ฃ+๐ฆ ๐๐ฃ/๐๐ฆ=(โ๐^๐ฃ (1 โ ๐ฃ))/(1 + ๐^๐ฃ ) ๐ฆ ๐๐ฃ/๐๐ฆ=(โ๐^๐ฃ (1 โ ๐ฃ))/(1 + ๐^๐ฃ )โ๐ฃ ๐ฆ ๐๐ฃ/๐๐ฆ=(โ๐^๐ฃ+ ๐ฃ๐^๐ฃ)/(1 + ๐^๐ฃ )โ๐ฃ ๐ฆ ๐๐ฃ/๐๐ฆ=(โ๐^๐ฃ+ ๐ฃ๐^๐ฃ โ ๐ฃ(1 + ๐^๐ฃ ))/(1 + ๐^๐ฃ ) ๐ฆ ๐๐ฃ/๐๐ฆ=(โ๐^๐ฃ+ ๐ฃ๐^๐ฃ โ ๐ฃ โ ๐ฃ๐^๐ฃ)/(1 + ๐^๐ฃ ) ๐ฆ ๐๐ฃ/๐๐ฆ=(โ๐^๐ฃโ ๐ฃ)/(1 + ๐^๐ฃ ) ๐ฆ ๐๐ฃ/๐๐ฆ=(โ(๐^๐ฃ+ ๐ฃ))/(1 + ๐^๐ฃ ) ๐ฆ ๐๐ฃ/๐๐ฆ=(โ(๐^๐ฃ+ ๐ฃ))/(1 + ๐^๐ฃ ) ใ1 + ๐ใ^๐ฃ/(๐ฃ + ๐^๐ฃ ) ๐๐ฃ = (โ๐๐ฆ)/๐ฆ Integrating both sides โซ1โใใ1 + ๐ใ^๐ฃ/(๐ฃ + ๐^๐ฃ ) ๐๐ฃ" " ใ =โซ1โ(โ๐๐ฆ)/๐ฆ โซ1โใใ1 + ๐ใ^๐ฃ/(๐ฃ + ๐^๐ฃ ) ๐๐ฃใ=โlogโกใ|๐ฆ|ใ+logโก๐ Putting v + ev = t (1 + ev) dv = dt Thus, our equation becomes โซ1โ๐๐ก/๐ก=โlogโกใ|๐ฆ|ใ+logโก๐ logโกใ|๐ก|ใ=โlogโกใ|๐ฆ|ใ+logโก๐ Putting back value of t = v + ev logโกใ|๐ฃ+๐^๐ฃ |ใ=โlogโกใ|๐ฆ|ใ+logโก๐ logโกใ|๐ฃ+๐^๐ฃ |ใ+logโกใ|๐ฆ|ใ=logโก๐ logโก(|๐ฃ+๐^๐ฃ |ร|๐ฆ|)=logโก๐ logโก((๐ฃ+๐^๐ฃ )ร๐ฆ)=logโก๐ logโก(๐ฃ๐ฆ+๐^๐ฃ ๐ฆ)=logโก๐ Putting back value of v = ๐ฅ/๐ฆ logโก(๐ฅ/๐ฆร๐ฆ+๐^(๐ฅ/๐ฆ) ๐ฆ)=logโก๐ logโก(๐ฅ+๐^(๐ฅ/๐ฆ) ๐ฆ)=logโก๐ Canceling log ๐+๐๐^(๐/๐)=๐ช