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


Transcript

Example 13 Show that the family of curves for which the slope of the tangent at any point (𝑥 , 𝑦) on its (𝑥^2+𝑦^2)/2𝑥𝑦 , is given by 𝑥^2−𝑦^2=𝑐𝑥 We know that the slope of the tangent at (𝑥 ,𝑦) of a curve is 𝑑𝑦/𝑑𝑥 Given slope of tangent at (𝑥 , 𝑦) is (𝑥^2 +〖 𝑦〗^2)/2𝑥𝑦 Therefore 𝒅𝒚/𝒅𝒙=(𝒙^𝟐 +〖 𝒚〗^𝟐)/𝟐𝒙𝒚 Step 1: Find 𝑑𝑦/𝑑𝑥 𝑑𝑦/𝑑𝑥=(𝑥^2 +𝑦^2)/2𝑥𝑦 Step 2: Put 𝑑𝑦/𝑑𝑥= F(𝑥 𝑦) So, F(𝑥 𝑦)=(𝑥^2 + 𝑦^2)/2𝑥𝑦 Finding F(𝜆𝑥 ,𝜆𝑦) F(𝜆𝑥 ,𝜆𝑦)=((𝜆𝑥)^2 + (𝜆𝑦)^2)/2(𝜆𝑥)(𝜆𝑦) =(𝜆^2 𝑥^2 + 𝜆^2 𝑦^2)/(2𝜆^2 𝑥𝑦) =(𝜆^2 (𝑥^2 +〖 𝑦〗^2 ))/(𝜆^2 2𝑥𝑦) = ((𝑥^2 +〖 𝑦〗^2 ))/( 2𝑥𝑦) = F(𝑥 , 𝑦) So , F(𝝀𝒙 ,𝝀𝒚)= F(𝑥 , 𝑦) =𝝀° "F" (𝒙 , 𝒚) So , "F" (𝑥 , 𝑦) is homogeneous function of degree zero, Therefore given equation is a homogeneous differential equation Step 3: Solving 𝑑𝑦/𝑑𝑥 by putting 𝑦=𝑣𝑥 𝑑𝑦/𝑑𝑥=(𝑥^2 + 𝑦^2)/2𝑥𝑦 Put 𝒚=𝒗𝒙 Diff. w.r.t. 𝑥 𝑑𝑦/𝑑𝑥=𝑑/𝑑𝑥 (𝑣𝑥) 𝑑𝑦/𝑑𝑥=𝑥 𝑑𝑣/𝑑𝑥+𝑣 𝑑𝑥/𝑑𝑥 𝒅𝒚/𝒅𝒙=𝒙𝒅𝒗/𝒅𝒙+𝒗 Putting values 𝑜𝑓 𝑑𝑦/𝑑𝑥 and y in (i") " v + 𝒙 𝒅𝒗/𝒅𝒙=(𝒙^𝟐 + (𝒗〖𝒙)〗^𝟐)/(𝟐𝒙 (𝒗𝒙)) " " 𝑥 𝑑𝑣/𝑑𝑥=(𝑥^2 + 𝑣^2 𝑥^2)/(2𝑥^2 𝑣) − 𝑣" " 𝑥 𝑑𝑣/𝑑𝑥=(𝑥^2 + 𝑣^2 𝑥^2 − 2𝑥^2 𝑣 . 𝑣)/(2𝑥^2 𝑣) 𝑥 𝑑𝑣/𝑑𝑥=(𝑥^2 + 𝑣^2 𝑥^2 − 2𝑥^2 𝑣^2)/(2𝑥^2 𝑣) 𝑥 𝑑𝑣/𝑑𝑥=(𝑥^2 − 𝑣^2 𝑥^2)/(2𝑥^2 𝑣) 𝑥 𝑑𝑣/𝑑𝑥=(𝑥^2 (1 − 𝑣^2 ))/(𝑥^2. 2𝑣) 𝑥 𝑑𝑣/𝑑𝑥=(1 − 𝑣^2)/2𝑣 𝑑𝑣/𝑑𝑥=(1 − 𝑣^2)/2𝑣 . 1/𝑥 (2𝑣 𝑑𝑣)/(1 − 𝑣^2 )=𝑑𝑥/𝑥 (2𝑣 𝑑𝑣)/(−(𝑣^2 − 1) )=𝑑𝑥/𝑥 (𝟐𝒗 𝒅𝒗 )/(𝒗^𝟐 − 𝟏)=(−𝒅𝒙)/𝒙 Integrating Both Sides ∫1▒〖(2𝑣 𝑑𝑣 )/(𝑣^2−1)=∫1▒(−𝑑𝑥)/𝑥〗 ∫1▒(𝟐𝒗 )/(𝒗^𝟐 − 𝟏) 𝒅𝒗=−𝒍𝒐𝒈|𝒙|+𝒄 Solving ∫1▒(𝟐𝒗 )/(𝒗^𝟐 − 𝟏) 𝒅𝒗 Put 𝒗^𝟐−𝟏=𝒕 Diff. w.r.t. 𝑣 𝑑(𝑣^2 −1)/𝑑𝑣=𝑑𝑡/𝑑𝑣 2𝑣=𝑑𝑡/𝑑𝑣 " " 𝑑𝑣=𝑑𝑡/2𝑣 " " ∫1▒𝟐𝒗/(𝒗^𝟐 −𝟏) 𝒅𝒗 = ∫1▒〖𝟐𝒗/𝒕 〗 𝒅𝒕/𝟐𝒗 =∫1▒𝑑𝑡/𝑡 = log |𝑡| Putting t = v2 – 1 = log |𝒗^𝟐−𝟏| From (2) ∫1▒(2𝑣 )/(𝑣^2 − 1) 𝑑𝑣=−𝑙𝑜𝑔|𝑥|+𝑐 "log " |𝑣^2−1|=−𝑙𝑜𝑔|𝑥|+𝑐1 Putting 𝑣𝑥=𝑦 or 𝑣=𝑦/𝑥 "log " |(𝒚/𝒙)^𝟐−𝟏|=−𝒍𝒐𝒈|𝒙|+𝒄𝟏 "log " |(𝑦/𝑥)^2−1|+𝑙𝑜𝑔|𝑥|=+𝑐1 "log" |[(𝑦/𝑥)^2−1]𝑥|=𝑐1 "log" |[𝑦^2/𝑥^2 −1]𝑥|=𝑐1 Putting 𝑐1=log⁡𝑐 𝑙𝑜𝑔|(𝑦^2/𝑥^2 −1)𝑥|=log⁡ 𝑐1 Removing log (𝒚^𝟐/𝒙^𝟐 −𝟏)𝒙=𝑪𝟏 (𝑥𝑦^2)/𝑥^2 −𝑥=𝐶1 𝑦^2/𝑥−𝑥=𝐶1 (𝑦^2−𝑥^2)/𝑥=𝐶1" " 𝑦^2−𝑥^2=𝐶1𝑥 𝑥^2−𝑦^2=−𝐶1𝑥 Put 𝑐=−𝐶1 𝒙^𝟐−𝒚^𝟐=𝒄𝒙 Hence Proved

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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.