Solve the differential equation: ye^(x/y) dx=(xe^(x/y)+y^2 )dy,(y≠0)
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CBSE Class 12 Sample Paper for 2024 Boards
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CBSE Class 12 Sample Paper for 2024 Boards
Last updated at Aug. 10, 2023 by Teachoo
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y𝑒^(𝑥/𝑦) 𝑑𝑥=(𝑥𝑒^(𝑥/𝑦)+𝑦^2 )𝑑𝑦 Step 1: Finding 𝑑𝑥/𝑑𝑦 y𝑒^(𝑥/𝑦) 𝑑𝑥=(𝑥𝑒^(𝑥/𝑦)+𝑦^2 )𝑑𝑦 𝒅𝒙/𝒅𝒚=(𝒙𝒆^(𝒙/𝒚) + 𝒚^𝟐)/(𝒚𝒆^(𝒙/𝒚) ) Step 2 : Solving 𝑑𝑥/𝑑𝑦 by Putting 𝑥=𝑣𝑦 𝑑𝑥/𝑑𝑦=(𝑥𝑒^(𝑥/𝑦) + 𝑦^2)/(𝑦𝑒^(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=(𝑥𝑒^(𝑥/𝑦)+𝑦^2)/(𝑦𝑒^(𝑥/𝑦) ) 𝒗+𝒚 𝒅𝒗/𝒅𝒚=(𝒗𝒚𝒆^𝒗+𝑦^2)/(𝒚𝒆^𝒗 ) 𝑣+𝑦 𝑑𝑣/𝑑𝑦=(𝑣〖𝑦𝑒〗^𝑣)/(𝑦𝑒^𝑣 ) + 𝑦^2/(𝑦𝑒^𝑣 ) v+𝑦 𝑑𝑣/𝑑𝑦=𝑣+ 𝑦/𝑒^𝑣 𝑦 𝑑𝑣/𝑑𝑦=𝑦/𝑒^𝑣 𝑑𝑣/𝑑𝑦=1/〖 𝑒〗^𝑣 〖 𝒆〗^𝒗 𝒅𝒗=𝒅𝒚 Integrating Both Sides ∫1▒〖〖 𝒆〗^𝒗 𝑑𝑣〗= ∫1▒𝑑𝑦 〖 𝒆〗^𝒗=𝒚+𝒄 Putting back 𝑣=𝑥/𝑦 𝒆^(𝒙/𝒚)=𝒚+𝒄