Example 17 - Show 2y e x/y dx + (y - 2x ex/y) dy = 0, particular - Examples

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 17 Show that the differential equation 2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥+ 𝑦−2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯𝑑𝑦=0 is homogeneous and find its particular solution , given that, 𝑥=0 when 𝑦=1 Step 1: Find 𝑑𝑥﷮𝑑𝑦﷯ 2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥+ 𝑦−2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯𝑑𝑦=0 2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥=− 𝑦−2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯𝑑𝑦 2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥= 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯−𝑦﷯𝑑𝑦 𝑑𝑥﷮𝑑𝑦﷯= 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ − 𝑦﷯﷮2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯ Step 2: Put F 𝑥 , 𝑦﷯= 𝑑𝑥﷮𝑑𝑦﷯ and find F 𝜆𝑥 ,𝜆𝑦﷯ F 𝑥 , 𝑦﷯= 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ − 𝑦﷮2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯ Finding F 𝜆𝑥 ,𝜆𝑦﷯ F 𝜆𝑥 ,𝜆𝑦﷯= 2 𝜆𝑥﷯ 𝑒﷮ 𝜆𝑥﷮𝜆𝑦﷯ −𝜆𝑦﷯﷮2𝜆𝑦 𝑒﷮ 𝜆𝑥﷮𝜆𝑦﷯ ﷯﷯ = 𝜆 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ − 𝑦﷯﷮𝜆 . 2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯ = 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ − 𝑦﷮2𝑦 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 𝑑𝑥﷮𝑑𝑦﷯= 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ − 𝑦﷮2𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯ Put 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﷯= 2𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ − 𝑦﷮2𝑦 𝑒﷮ 𝑥 ﷮𝑦﷯﷯﷯ 𝑣+𝑦 𝑑𝑣﷮𝑑𝑦﷯= 2𝑣 𝑒﷮𝑣﷯ −1﷮2 𝑒﷮𝑣﷯ ﷯ 𝑦 𝑑𝑣﷮𝑑𝑦﷯= 2𝑣 𝑒﷮𝑣﷯ −1﷮2 𝑒﷮𝑣﷯ ﷯−𝑣 𝑦 𝑑𝑣﷮𝑑𝑦﷯= 2 𝑣 𝑒﷮𝑣﷯ −1 − 2 𝑣 𝑒﷮𝑣﷯﷮2 𝑒﷮𝑣﷯﷯ 𝑦 𝑑𝑣﷮𝑑𝑦﷯= −1﷮2 𝑒﷮𝑣﷯﷯ 𝑦 𝑑𝑣﷮𝑑𝑦﷯= −1﷮2 𝑒﷮𝑣﷯﷯ 2 𝑒﷮𝑣﷯ 𝑑𝑣= −𝑑𝑦﷮ 𝑦﷯ Integrating Both Sides ﷮﷮2 𝑒﷮𝑣﷯ 𝑑𝑣﷯= ﷮﷮ −𝑑𝑦﷮ 𝑦﷯﷯ 2 𝑒﷮𝑣﷯=− log﷮ 𝑦﷯﷯+𝑐 Putting 𝑣= 𝑥﷮𝑦﷯ 2 𝑒﷮ 𝑥﷮𝑦﷯﷯=−𝑙𝑜𝑔 𝑦﷯+𝑐 2 𝑒﷮ 𝑥﷮𝑦﷯﷯+𝑙𝑜𝑔 𝑦﷯=𝑐 Given that at 𝑥=0 , 𝑦=1 Putting 𝑥=0 and 𝑦=1 in (2) 2 𝑒﷮ 0﷮1﷯﷯−𝑙𝑜𝑔 1﷯=𝑐 2 ×1+0=𝑐 𝑐=2 Put Value of 𝑐 in (2) i.e., 2 𝑒﷮ 𝑥﷮𝑦﷯﷯+𝑙𝑜𝑔 𝑦﷯=𝐶 2 𝑒﷮ 𝑥﷮𝑦﷯﷯ + log 𝑦﷯ = c 𝟐 𝒆﷮ 𝒙﷮𝒚﷯﷯ + log 𝒚﷯ = 2 is the particular solution of given differential equation

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