Question 10 - Formation of Differntial equation when general solution given - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
Formation of Differntial equation when general solution given
Formation of Differntial equation when general solution given
Last updated at April 16, 2024 by Teachoo
Question 10 Form the differential equation of the family of circle having a center on y-axis and radius 3 units. General equation of circle is :- (𝑥−𝑎)^2+(𝑦−𝑏)^2=𝑟^2 Given center is on y-axis ∴ Center = (0, b) And, Radius = 3 Hence, our equation becomes (𝑥−0)^2+(𝑦−𝑏)^2=(3)^2 𝑥^2+(𝑦−𝑏)^2=9 Differentiating Both Sides w.r.t. 𝑥 2𝑥+2(𝑦−𝑏)[𝑑𝑦/𝑑𝑥−0]=0 2𝑥+2(𝑦−𝑏)𝑦′=0 2[𝑥+(𝑦−𝑏)𝑦′]=0 𝑥+(𝑦−𝑏) 𝑦^′=0 (𝑦−𝑏)𝑦′=−𝑥 (𝑦−𝑏)= (−𝑥)/𝑦^′ Putting the value of (𝑦−𝑏) in equation (1) x2 + (y − b)2 = 9 𝑥^2+[(−𝑥)/𝑦^′ ]^2=9 𝑥^2+𝑥^2/〖𝑦^′〗^2 =9 (𝑥^2 〖𝑦^′〗^2+ 𝑥^2)/〖𝑦^′〗^2 =9 𝑥^2 〖𝑦^′〗^2+𝑥^2=9〖𝑦^′〗^2 𝑥^2 〖𝑦^′〗^2−9〖𝑦^′〗^2+𝑥^2=0 〖𝑦^′〗^2 (𝑥^2−9)+𝑥^2=0 ∴ (𝒙^𝟐−𝟗) 〖𝒚^′〗^𝟐+𝒙^𝟐=𝟎