Question 2 - 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 2 Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes . Letβs first draw the figure Let C be the family of circles in first quadrant touching coordinate axes Let radius be π β΄ Center of circle = (βπ, π) Thus, Equation of a circle is (π₯βπ)^2+(π¦βπ)^2=π^2 π₯^2+π^2β2ππ₯+π¦^2+π^2β2ππ¦=π^2 π₯^2+π¦^2β2ππ₯β2ππ¦+π^2=0 Since there is only one variable, we differentiate once Differentiating w.r.t. π₯ (π₯^2 + π¦^2 β 2ππ₯ β 2ππ¦+π^2 )^β²=(0)^β² 2π₯+2π¦π¦β²β2πβ2ππ¦β²+0=0 π₯+π¦π¦β²βπβππ¦β²=0 π₯+π¦π¦^β²=π+ππ¦^β² π₯+π¦π¦^β²=π(1+π¦^β²) π = (π₯ + π¦π¦^β²)/(1 + π¦^β² ) Putting Value of π in (π₯βπ)^2+(π¦βπ)^2=π^2 (π₯β[(π₯ + π¦ π¦^β²)/(1 + π¦^β² )])^2+(π¦β[(π₯ + π¦ π¦^β²)/(1 + π¦^β² )])^2=((π₯ + π¦π¦^β²)/(1 + π¦^β² ))^2 (π₯(1 + π¦^β² ) β (π₯ + π¦π¦^β² ))^2/(1 + π¦^β² )^2 +(π¦(1 + π¦^β² ) β (π₯ + π¦π¦^β² ))^2/(1 + π¦^β² )^2 =(π₯ + π¦π¦^β² )^2/(1 + π¦^β² )^2 (π₯+π₯π¦^β²βπ₯βπ¦π¦^β² )^2+(π¦+π¦π¦^β²βπ₯βπ¦π¦^β² )^2=(π₯+π¦π¦^β² )^2 (π₯π¦^β²βπ¦π¦^β² )^2+(π¦βπ₯)^2=(π₯+π¦π¦^β² )^2 (π₯βπ¦)^2 (π¦^β² )^2+(π₯βπ¦)^2=(π₯+π¦π¦^β² )^2 (π₯βπ¦)^2 (1+γπ¦^β²γ^2 )=(π₯+π¦π¦^β² )^2 (π+ππ^β² )^π=(πβπ)^π (π+γπ^β²γ^π )