Example 4 - Form differential equation y = mx - Class 12 - Formation of Differntial equation when general solution given

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  1. Chapter 9 Class 12 Differential Equations
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Example 4 Form the differential equation representing the family of curves ๐‘ฆ=๐‘š๐‘ฅ, where ; ๐‘š is arbitrary constant . ๐‘ฆ=๐‘š๐‘ฅ Step 1: Calculate ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ= ๐‘‘ ๐‘š๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ=๐‘š ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ=๐‘š or ๐‘š= ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ Now, ๐‘ฆ=๐‘š๐‘ฅ Putting ๐‘š= ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ from (1) ๐‘ฆ= ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ๐‘ฅ x ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏโˆ’๐‘ฆ=0 โˆด Differential Equation is ๐’™ ๐’…๐’š๏ทฎ๐’…๐’™๏ทฏโˆ’๐’š=๐ŸŽ

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