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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 7 Form the differential equation of the family of circles touching the x-axis at origin. We know that, Equation of Circle is (๐‘ฅโˆ’๐‘Ž)^2+(๐‘ฆโˆ’๐‘)^2=๐‘Ÿ^2 Center =(๐‘Ž,๐‘) Radius = ๐‘Ÿ Since the circle touches the x-axis at origin The center will be on the y-axis So, x-coordinate of center is 0 i.e. a = 0 โˆด Center = (0 , ๐‘) And, ๐‘Ÿ๐‘Ž๐‘‘๐‘–๐‘ข๐‘ =๐‘ So, Equation of Circle (๐‘ฅโˆ’0)^2+(๐‘ฆโˆ’๐‘)^2=๐‘^2 ๐‘ฅ^2+(๐‘ฆโˆ’๐‘)^2=๐‘^2 ๐‘ฅ^2+๐‘ฆ^2โˆ’2๐‘ฆ๐‘+๐‘^2=๐‘^2 ๐‘ฅ^2+๐‘ฆ^2โˆ’2๐‘ฆ๐‘=0 ๐‘ฅ^2+๐‘ฆ^2=2๐‘ฆ๐‘ Since there is one variable b, we differentiate once Diff. w.r.t. ๐‘ฅ 2๐‘ฅ+2๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=2๐‘.๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ๐‘ฅ+๐‘ฆ๐‘ฆ^โ€ฒ=๐‘.๐‘ฆโ€ฒ [(๐‘ฅ + ๐‘ฆ๐‘ฆ^โ€ฒ)/๐‘ฆ^โ€ฒ ]=๐‘ Putting Value of ๐‘ in (1) ๐‘ฅ^2+๐‘ฆ^2=2๐‘ฆ[(๐‘ฅ + ๐‘ฆ๐‘ฆ^โ€ฒ)/๐‘ฆ^โ€ฒ ] (๐‘ฅ^2+๐‘ฆ^2 ) ๐‘ฆ^โ€ฒ=2๐‘ฆ(๐‘ฅ+๐‘ฆ๐‘ฆ^โ€ฒ ) (๐‘ฅ^2+๐‘ฆ^2 ) ๐‘ฆ^โ€ฒ=2๐‘ฆ๐‘ฅ+2๐‘ฆ^2 ๐‘ฆ^โ€ฒ ๐‘ฅ^2 ๐‘ฆ^โ€ฒ+๐‘ฆ^2 ๐‘ฆ^โ€ฒ=2๐‘ฆ๐‘ฅ+2๐‘ฆ^2 ๐‘ฆ^โ€ฒ ๐‘ฅ^2 ๐‘ฆ^โ€ฒ+๐‘ฆ^2 ๐‘ฆ^โ€ฒโˆ’2๐‘ฆ^2 ๐‘ฆ^โ€ฒ=2๐‘ฆ๐‘ฅ ๐‘ฅ^2 ๐‘ฆ^โ€ฒโˆ’๐‘ฆ^2 ๐‘ฆ^โ€ฒ=2๐‘ฆ๐‘ฅ ๐‘ฆ^โ€ฒ (๐‘ฅ^2โˆ’๐‘ฆ^2 )=2๐‘ฅ๐‘ฆ ๐ฒโ€ฒ=๐Ÿ๐’™๐’š/(๐’™^๐Ÿ โˆ’ ๐’š^๐Ÿ )

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.