Question

Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $Py"+Q{y}^{\prime }+1=0$, where $P$, $Q$ are functions of $x,y$ and $y^{\prime }$ (here $y^{\prime }=\frac{dy}{dx},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}$), then which of the following statements is (are) true ?

• A. $P=y+x$
• B. $P=y-x$
• C. $P+Q=1-x+y+y^{\prime }+(y^{\prime }{)}^2$
• D. $P-Q=x+y-y^{\prime }-(y^{\prime }{)}^2$

Answer 1

Answer: (B), (C)

Let the equation of circle is
${\left(x-\alpha \right)}^2+{\left(y-\alpha \right)}^2=r^2$
$\Rightarrow x^2+y^2-2\alpha x-2\alpha y+2{\alpha }^2-r^2=0$
differentiate w.r.t. x
$\Rightarrow 2x+2y{y}^{\prime }-2\alpha -2\alpha y^{\prime }=0$
$\Rightarrow \alpha =\frac{x+y{y}^{\prime }}{1+y^{\prime }}...\left(i\right)$
differentiate again w.r.t. x
$2+2{\left(y^{\prime }\right)}^2+2yy-2\alpha y=0$
$\Rightarrow \alpha =\frac{1+{\left(y^{\prime }\right)}^2+y{y}^{\prime \prime }}{y^{\prime \prime }}...\left(ii\right)$
from $\left(i\right)\&\left(ii\right)$
$xy+y{y}^{\prime }y=1+{\left(y^{\prime }\right)}^2+yy+y^{\prime }+{\left(y^{\prime }\right)}^3+y{y}^{\prime }{y}^{\prime \prime }$
$\Rightarrow \left(y-x\right)y+y^{\prime }\left[y^{\prime }+1+{\left(y^{\prime }\right)}^2\right]+1=0$
$P=y-x$
$Q=y^{\prime }+1+{\left(y^{\prime }\right)}^2$