1. Chapter 9 Class 12 Differential Equations
2. Serial order wise

Transcript

Ex 9.3, 4 Form a differential equation representing the given family of curves by eliminating arbitrary constants ๐ and ๐. ๐ฆ= ๐๏ทฎ2๐ฅ๏ทฏ ๐+๐๐ฅ๏ทฏ ๐ฆ= ๐๏ทฎ2๐ฅ๏ทฏ ๐+๐๐ฅ๏ทฏ โด Differentiating Both Sides w.r.t. ๐ฅ ๐ฆ๏ทฎโฒ๏ทฏ= ๐๏ทฎ๐๐ฅ๏ทฏ ๐๏ทฎ2๐ฅ๏ทฏ ๐+๐๐ฅ๏ทฏ๏ทฏ ๐ฆ๏ทฎโฒ๏ทฏ= ๐ ๐๏ทฎ2๐ฅ๏ทฏ๏ทฏ๏ทฎ๐๐ฅ๏ทฏ. ๐+๐๐ฅ๏ทฏ+ ๐๏ทฎ2๐ฅ ๏ทฏ ๐ ๐ + ๐๐ฅ๏ทฏ๏ทฎ๐๐ฅ๏ทฏ ๐ฆ๏ทฎโฒ๏ทฏ= 2๐๏ทฎ2๐ฅ๏ทฏ ๐+๐๐ฅ๏ทฏ+ ๐๏ทฎ2๐ฅ๏ทฏ.๐ ๐๏ทฎโฒ๏ทฏ= ๐๏ทฎ๐๐๏ทฏ ๐๐+๐๐๐+๐๏ทฏ Again differentiating w.r.t.x ๐ฆ^โฒ=๐/๐๐ฅ (๐^2๐ฅ [2๐+2๐๐ฅ+๐]) yโ = (๐ (๐^2๐ฅ))/๐๐ฅ [2๐+2๐๐ฅ+๐]+๐^2๐ฅ (๐ [2๐+2๐๐ฅ+๐])/๐๐ฅ yโ = 2๐^2๐ฅ [2๐+2๐๐ฅ+๐]+๐^2๐ฅร2๐ Putting yโ=๐^2๐ฅ [2๐+2๐๐ฅ+๐] yโ = 2yโ + ๐^2๐ฅร2๐ yโ = 2yโ + 2๐^2๐ฅ ๐ yโ โ 2yโ = 2๐^2๐ฅ ๐ Also, yโ โ 2y = ๐^2๐ฅ [2๐+2๐๐ฅ +๐]โ2๐^2๐ฅ (๐+๐๐ฅ) yโ โ 2y = 2a๐^2๐ฅ+2๐๐ฅ ๐^2๐ฅ+๐^2๐ฅ ๐โ2๐ใ ๐ใ^2๐ฅโ2๐๐ฅ ๐^2๐ฅ yโ โ 2y = (2๐ใ ๐ใ^2๐ฅโ2๐ใ ๐ใ^2๐ฅ )+(2๐๐ฅ ๐^2๐ฅโ2๐๐ฅ ๐^2๐ฅ )+๐^2๐ฅ ๐ yโ โ 2y = 0 + 0 + ๐^2๐ฅ ๐ yโ โ 2y = ๐^2๐ฅ ๐ Now ((1))/((2)) , (๐ฆ" โ 2๐ฆ)/(๐ฆ^(โฒ ) โ 2๐ฆ)=(2๐^2๐ฅ ๐)/(๐^2๐ฅ ๐) (๐ฆ^โฒโฒ โ 2๐ฆ^โฒ)/(๐ฆ^โฒโ2๐ฆ)= 2 yโ โ 2yโ = 2(yโ โ 2y) yโ โ 2yโ = 2yโ โ 4y yโ โ 2yโ โ 2yโ โ 4y = 0 yโ โ 4yโ โ 4y = 0