Question

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?

Answer 1

`(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`

Answer 2

`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`