Example 5 - Form differential equation y = a sin (x + b) - Examples


  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 5 Form the differential equation representing the family of curves ๐‘ฆ=๐‘Ž sin๏ทฎ ๐‘ฅ+๐‘๏ทฏ, ๏ทฏ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘Ž,๐‘ are arbitrary constants. The numbers of constants, is equal to the number of time we differentiate Here, there are two constants , So, we differentiate twice ๐‘ฆ=๐‘Ž ๐‘ ๐‘–๐‘› ๐‘ฅ+๐‘๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ= ๐‘‘ ๐‘Ž ๐‘ ๐‘–๐‘› ๐‘ฅ + ๐‘๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ=๐‘Ž ๐‘๐‘œ๐‘  ๐‘ฅ+๐‘๏ทฏ Now, ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ๐‘‘ ๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ= ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ๏ทฏ = ๐‘‘ ๐‘Ž ๐‘๐‘œ๐‘  ๐‘ฅ + ๐‘๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ =โˆ’a sin ๐‘ฅ+๐‘๏ทฏ โˆด ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ๐‘‘ ๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ=โˆ’๐‘Ž ๐‘ ๐‘–๐‘› ๐‘ฅ+๐‘๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘ ๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ = โˆ’y ๐‘‘๏ทฎ2๏ทฏ๐‘ฆ๏ทฎ๐‘‘ ๐‘ฅ๏ทฎ2๏ทฏ๏ทฏ+๐‘ฆ=0 Hence, our differential Equation is ๐’…๏ทฎ๐Ÿ๏ทฏ๐’š๏ทฎ๐’… ๐’™๏ทฎ๐Ÿ๏ทฏ๏ทฏ+๐’š=๐ŸŽ

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