Slide1.JPG

Slide2.JPG
Slide3.JPG Slide4.JPG Slide5.JPG Slide6.JPG

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 9.4, 6 Show that the given differential equation is homogeneous and solve each of them. 𝑥 𝑑𝑦−𝑦 𝑑𝑥=√(𝑥^2+𝑦^2 ) 𝑑𝑥 Step 1: Find 𝑑𝑦/𝑑𝑥 x dy − y dx = √(𝑥^2+𝑦^2 ) dx x dy = √(𝑥^2+𝑦^2 ) dx + y dx x dy = (√(𝑥^2+𝑦^2 )+𝑦) dx 𝒅𝒚/𝒅𝒙 = (√(𝒙^𝟐 + 𝒚^𝟐 ) + 𝒚)/𝒙 Step 2: Put 𝑑𝑦/𝑑𝑥 = F(x, y) and find F(𝜆x, 𝜆y) F(x, y) = 𝑑𝑦/𝑑𝑥 = (√(𝑥^(2 )+ 𝑦^2 ) + 𝑦)/𝑥 F(𝜆 x, 𝜆y) = (√(〖(𝜆𝑥)〗^2 + (𝜆^2 𝑦^2 ) )+ 𝜆𝑦)/𝜆𝑥 = (√(𝜆^2 𝑥^2 + 𝜆^2 𝑦^2 ) + 𝜆𝑦)/𝜆𝑥 = (√(𝜆^2 (𝑥^2 + 𝑦^2)) + 𝜆𝑦)/𝜆𝑥= (𝜆√(𝑥^2 + 𝑦^2 ) + 𝜆𝑦)/𝜆𝑥 = (√(𝑥^2 + 𝑦^2 ) + 𝑦)/𝑥 = F(x, y) Hence, F(𝜆x, 𝜆y) = F(x, y) = 𝜆° F(x, y) Hence, F(x, y) is a homogenous Function of with degree 0 So, 𝑑𝑦/𝑑𝑥 is a homogenous differential equation. Step 3 - Solving 𝑑𝑦/𝑑𝑥 by putting y = vx Putting y = vx. Differentiating w.r.t.x 𝑑𝑦/𝑑𝑥 = 𝑥 𝑑𝑣/𝑑𝑥+𝑣 𝑑𝑥/𝑑𝑥 𝒅𝒚/𝒅𝒙 = 𝒙 𝒅𝒗/𝒅𝒙 + 𝒗 Putting value of 𝑑𝑦/𝑑𝑥 and y = vx in (1) 𝑑𝑦/𝑑𝑥=(√(𝑥^2 + 𝑦^2 )+ 𝑦)/𝑥 x 𝑑𝑣/𝑑𝑥+𝑣=(√(𝑥^2 + (𝑣𝑥)^2 ) + (𝑣𝑥))/𝑥 x 𝑑𝑣/𝑑𝑥+𝑣=(√(𝑥^2 + 𝑥^2 𝑣^2 ) + 𝑣𝑥)/𝑥 x 𝑑𝑣/𝑑𝑥+𝑣 =(√(𝑥^2 (1 + 𝑣^2)) + 𝑣𝑥)/𝑥 x 𝑑𝑣/𝑑𝑥+𝑣 =(𝑥√(1 + 𝑣^2 ) + 𝑣𝑥)/𝑥 x 𝑑𝑣/𝑑𝑥+𝑣 =(𝑥(√(1 + 𝑣^2 ) + 𝑣))/𝑥 x 𝒅𝒗/𝒅𝒙+𝒗= √(𝟏+𝒗^𝟐 )+𝒗 x 𝑑𝑣/𝑑𝑥= √(1+𝑣^2 )+𝑣 − 𝑣 x 𝑑𝑣/𝑑𝑥= √(1+𝑣^2 ) 𝑑𝑣/𝑑𝑥= √(1 + 𝑣^2 )/𝑥 𝒅𝒗/√(𝟏 + 𝒗^𝟐 )= 𝒅𝒙/𝒙 Integrating both sides. ∫1▒𝑑𝑣/√(1 + 𝑣^2 ) = ∫1▒𝑑𝑥/𝑥 ∫1▒𝒅𝒗/√(𝟏 + 𝒗^𝟐 ) = log |𝒙|+𝒄 We know that ∫1▒𝑑𝑣/√(𝑎^2 + 𝑥^2 ) =𝑙𝑜𝑔|𝑥+√(𝑥^2+𝑎^2 )|+𝑐 Putting a = 1, x = v log |𝑣+√(𝑣^2+1)| =𝑙𝑜𝑔|𝑥|+𝑐 log |𝑣+√(𝑣^2+1)| =𝑙𝑜𝑔|𝑐𝑥| v + √(𝒗^𝟐+𝟏) = cx Putting v = 𝑦/𝑥 𝒚/𝒙+√((𝒚/𝒙)^𝟐+𝟏)=𝒄𝒙 𝑦/𝑥+√(𝑦^2/𝑥^2 +1)=𝑐𝑥 𝑦/𝑥+√((𝑦^2 + 𝑥^2)/𝑥^2 )=𝑐𝑥 𝑦/𝑥+√(𝑦^2 + 𝑥^2 )/𝑥=𝑐𝑥 𝒚+√(𝒚^𝟐 〖+ 𝒙〗^𝟐 ) =𝒄𝒙^𝟐 ∴ General solution is 𝒚+√(𝒚^𝟐 〖+ 𝒙〗^𝟐 ) =𝒄𝒙^𝟐

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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.