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Ex 9.5, 10 - Show homogeneous: (1 + ex/y) dx + e x/y (1 - x/y)

Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations - Part 3 Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations - Part 4 Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations - Part 5 Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations - Part 6 Ex 9.5, 10 - Chapter 9 Class 12 Differential Equations - Part 7

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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+𝑒^(𝑥/𝑦) )𝑑𝑥+𝑒^(𝑥/𝑦) (1−𝑥/𝑦)𝑑𝑦=0 Since the equation is in the form 𝑥/𝑦 , we will take 𝑑𝑥/𝑑𝑦 Instead of 𝑑𝑦/𝑑𝑥 Step 1 : Find 𝑑𝑥/𝑑𝑦 (1+𝑒^(𝑥/𝑦) )𝑑𝑥+𝑒^(𝑥/𝑦) (1−𝑥/𝑦)𝑑𝑦 = 0 (1+𝑒^(𝑥/𝑦) ) dx = −𝑒^(𝑥/𝑦) (1−𝑥/𝑦)𝑑𝑦 𝑑𝑥/𝑑𝑦 = (−𝑒^(𝑥/𝑦) (1 − 𝑥/𝑦) )/(1 + 𝑒^(𝑥/𝑦) ) …(1) Step 2: Put 𝑑𝑥/𝑑𝑦 = F(x, y) and find F(𝜆x, 𝜆y) F(x, y) = (−𝑒^(𝑥/𝑦) (1 − 𝑥/𝑦) )/(1 + 𝑒^(𝑥/𝑦) ) F(𝜆x, 𝜆y) = (−𝑒^(𝜆𝑥/𝜆𝑦) (1 − 𝜆𝑥/𝜆𝑦) )/(1 + 𝑒^(𝜆𝑥/𝜆𝑦) ) = (−𝑒^(𝑥/𝑦) (1 − 𝑥/𝑦) )/(1 + 𝑒^(𝑥/𝑦) ) = 𝐹(𝑥, 𝑦) So, F(𝜆𝑥 ,𝜆𝑦)= F(𝑥 , 𝑦) = 𝜆° F(𝑥 , 𝑦) Thus , F(𝑥 ,𝑦) is a homogeneous function of degree zero Therefore given differential equation is homogeneous differential equation Step 3: Solving 𝑑𝑥/𝑑𝑦 by Putting 𝑥=𝑣𝑦 Putting 𝑥=𝑣𝑦 Diff. w.r.t. 𝑦 𝑑𝑥/𝑑𝑦=𝑑/𝑑𝑦 (𝑣𝑦) 𝑑𝑥/𝑑𝑦=𝑦 . 𝑑𝑣/𝑑𝑦+𝑣 𝑑𝑦/𝑑𝑦 𝑑𝑥/𝑑𝑦=𝑦 . 𝑑𝑣/𝑑𝑦+𝑣 Putting values of 𝑑𝑥/𝑑𝑦 and x in (1) 𝑑𝑥/𝑑𝑦=(−𝑒^(𝑥/𝑦) (1 − 𝑥/𝑦) )/(1 + 𝑒^(𝑥/𝑦) ) 𝑣+𝑦 𝑑𝑣/𝑑𝑦=(−𝑒^𝑣 (1 − 𝑣))/(1 + 𝑒^𝑣 ) 𝑦 𝑑𝑣/𝑑𝑦=(−𝑒^𝑣 (1 − 𝑣))/(1 + 𝑒^𝑣 )−𝑣 𝑦 𝑑𝑣/𝑑𝑦=(−𝑒^𝑣+ 𝑣𝑒^𝑣)/(1 + 𝑒^𝑣 )−𝑣 𝑦 𝑑𝑣/𝑑𝑦=(−𝑒^𝑣+ 𝑣𝑒^𝑣 − 𝑣(1 + 𝑒^𝑣 ))/(1 + 𝑒^𝑣 ) 𝑦 𝑑𝑣/𝑑𝑦=(−𝑒^𝑣+ 𝑣𝑒^𝑣 − 𝑣 − 𝑣𝑒^𝑣)/(1 + 𝑒^𝑣 ) 𝑦 𝑑𝑣/𝑑𝑦=(−𝑒^𝑣− 𝑣)/(1 + 𝑒^𝑣 ) 𝑦 𝑑𝑣/𝑑𝑦=(−(𝑒^𝑣+ 𝑣))/(1 + 𝑒^𝑣 ) 𝑦 𝑑𝑣/𝑑𝑦=(−(𝑒^𝑣+ 𝑣))/(1 + 𝑒^𝑣 ) 〖1 + 𝑒〗^𝑣/(𝑣 + 𝑒^𝑣 ) 𝑑𝑣 = (−𝑑𝑦)/𝑦 Integrating both sides ∫1▒〖〖1 + 𝑒〗^𝑣/(𝑣 + 𝑒^𝑣 ) 𝑑𝑣" " 〗 =∫1▒(−𝑑𝑦)/𝑦 ∫1▒〖〖1 + 𝑒〗^𝑣/(𝑣 + 𝑒^𝑣 ) 𝑑𝑣〗=−log⁡〖|𝑦|〗+log⁡𝑐 Putting v + ev = t (1 + ev) dv = dt Thus, our equation becomes ∫1▒𝑑𝑡/𝑡=−log⁡〖|𝑦|〗+log⁡𝑐 log⁡〖|𝑡|〗=−log⁡〖|𝑦|〗+log⁡𝑐 Putting back value of t = v + ev log⁡〖|𝑣+𝑒^𝑣 |〗=−log⁡〖|𝑦|〗+log⁡𝑐 log⁡〖|𝑣+𝑒^𝑣 |〗+log⁡〖|𝑦|〗=log⁡𝑐 log⁡(|𝑣+𝑒^𝑣 |×|𝑦|)=log⁡𝑐 log⁡((𝑣+𝑒^𝑣 )×𝑦)=log⁡𝑐 log⁡(𝑣𝑦+𝑒^𝑣 𝑦)=log⁡𝑐 Putting back value of v = 𝑥/𝑦 log⁡(𝑥/𝑦×𝑦+𝑒^(𝑥/𝑦) 𝑦)=log⁡𝑐 log⁡(𝑥+𝑒^(𝑥/𝑦) 𝑦)=log⁡𝑐 Canceling log 𝒙+𝒚𝒆^(𝒙/𝒚)=𝑪

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.