UPSC 2017 Maths Optional Paper 1 Q5a — Step-by-Step Solution

10 marks · Section B

Formulation of differential equations · ODEs · asked 2× in 14 yrs · Read the full method →

Question

Find the differential equation representing all the circles in the $x$ - $y$ plane.

A circle has 3 parameters $\Rightarrow$ differentiate three times and eliminate $g,f,c$ .

The general circle in the plane is

x^2+y^2+2gx+2fy+c=0,

with three arbitrary constants $g,f,c$ . To eliminate them we differentiate three times, producing a third-order ODE.

Differentiate once with respect to $x$ (treating $y=y(x)$ ):

2x+2yy'+2g+2fy'=0\quad\Longrightarrow\quad x+yy'+g+fy'=0.\tag{1}

Differentiate (1):

1+y'^2+yy''+fy''=0\quad\Longrightarrow\quad 1+y'^2+yy''+fy''=0.\tag{2}

Solve for $f$ :

f=-\frac{1+y'^2+yy''}{y''}.

Differentiate (2):

2y'y''+y'y''+yy'''+fy'''=0\quad\Longrightarrow\quad 3y'y''+yy'''+fy'''=0.\tag{3}

So $f=-\dfrac{3y'y''+yy'''}{y'''}$ . Equate the two expressions for $f$ :

\frac{1+y'^2+yy''}{y''}=\frac{3y'y''+yy'''}{y'''}.

Cross-multiplying,

(1+y'^2+yy'')\,y'''=(3y'y''+yy''')\,y''.

The $yy''y'''$ terms cancel on both sides:

(1+y'^2)\,y'''+yy''y'''=3y'y''^2+yy''y''',

(1+y'^2)\,y'''=3y'\,y''^2.

\boxed{\;(1+y'^2)\,\frac{d^3y}{dx^3}=3\,\frac{dy}{dx}\left(\frac{d^2y}{dx^2}\right)^2.\;}