Find the differential equation of the family of curves #color(white)(.)x^2+y^2+2ax+2by+c=0#,where #a,b,c# are parameters?

2 Answers
Jul 2, 2018

Answer:

#(y+b)(d^2y)/(dx^2)+((dy)/(dx))^2+1=0#

Explanation:

Differentiating #x^2+y^2+2ax+2by+c=0#

#2x+2y(dy)/(dx)+2a+2b(dy)/(dx)=0#

or #x+y(dy)/(dx)+a+b(dy)/(dx)=0# or #(dy)/(dx)=-(x+a)/(y+b)#

or #1+((dy)/(dx))^2+y(d^2y)/(dx^2)+b(d^2y)/(dx^2)=0#

i.e. #1+((dy)/(dx))^2+(y+b)(d^2y)/(dx^2)=0#

or #(y+b)(d^2y)/(dx^2)+((dy)/(dx))^2+1=0#

Jul 3, 2018

Answer:

# y_(x x x) + y_(x)^2y_(x x x) - 3y_(x)y_(x x)^2 = 0 #

Explanation:

#x^2+y^2+2ax+2by+c=0#

Taking 1st, 2nd, 3rd derivatives:

1st: #qquad 2 x + 2y y_(x)+2a +2by_(x) =0#

2nd: #qquad 2 + 2y_(x)^2 + 2 yy_(x x) +2by_(x x) =0 qquad square#

3rd: #qquad 6y_(x)y_(x x) + 2 y y_(x x x) +2by_(x x x) =0 qquad triangle#

  • #square * y_(x x x) - triangle * y_(x x)# to eliminate remaining #b# reference

# implies 2y_(x x x) + 2y_(x)^2y_(x x x) + cancel(2 yy_(x x)y_(x x x)) - 6y_(x)y_(x x)^2 - cancel(2 y y_(x x) y_(x x x) ) = 0 #

# :. y_(x x x) + y_(x)^2y_(x x x) - 3y_(x)y_(x x)^2 = 0 #