Examples

Chapter 9 Class 12 Differential Equations
Serial order wise

### Transcript

Question 4 Form the differential equation of the family of circles touching the x-axis at origin. We know that, Equation of Circle is (𝑥−𝑎)^2+(𝑦−𝑏)^2=𝑟^2 Center =(𝑎,𝑏) Radius = 𝑟 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 ∴ Center = (0 , 𝑏) And, 𝑟𝑎𝑑𝑖𝑢𝑠=𝑏 So, Equation of Circle (𝑥−0)^2+(𝑦−𝑏)^2=𝑏^2 𝑥^2+(𝑦−𝑏)^2=𝑏^2 𝑥^2+𝑦^2−2𝑦𝑏+𝑏^2=𝑏^2 𝑥^2+𝑦^2−2𝑦𝑏=0 𝑥^2+𝑦^2=2𝑦𝑏 Since there is one variable b, we differentiate once Diff. w.r.t. 𝑥 2𝑥+2𝑦 𝑑𝑦/𝑑𝑥=2𝑏.𝑑𝑦/𝑑𝑥 𝑥+𝑦𝑦^′=𝑏.𝑦′ [(𝑥 + 𝑦𝑦^′)/𝑦^′ ]=𝑏 Putting Value of 𝑏 in (1) 𝑥^2+𝑦^2=2𝑦[(𝑥 + 𝑦𝑦^′)/𝑦^′ ] (𝑥^2+𝑦^2 ) 𝑦^′=2𝑦(𝑥+𝑦𝑦^′ ) (𝑥^2+𝑦^2 ) 𝑦^′=2𝑦𝑥+2𝑦^2 𝑦^′ 𝑥^2 𝑦^′+𝑦^2 𝑦^′=2𝑦𝑥+2𝑦^2 𝑦^′ 𝑥^2 𝑦^′+𝑦^2 𝑦^′−2𝑦^2 𝑦^′=2𝑦𝑥 𝑥^2 𝑦^′−𝑦^2 𝑦^′=2𝑦𝑥 𝑦^′ (𝑥^2−𝑦^2 )=2𝑥𝑦 𝐲′=𝟐𝒙𝒚/(𝒙^𝟐 − 𝒚^𝟐 )

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.