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Home  >>  CBSE XII  >>  Math  >>  Differential Equations
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The differential equation $y\Large \frac{dy}{dx}\normalsize+x=c$ represents:\begin{array}{1 1}(A)\;family \;of \;hyperbolas & (B)\;family \;of \;parabolas \\(C)\;family \;of \;ellipses & (D)\;family\; of \;circles\end{array}

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  • If a linear differential equation is of the type $\large \frac{dy}{dx}$$=f(x)$ then it can be solved by seperating the variables and then integrating
Given $ y \large\frac{dy}{dx}$$+x=c$
This can be written as $y. \large \frac{dy}{dx}$$=-x+c$
Now seperating the variables we get,
$y dy=(-x+c)dx$
Integrating on both sides we get,
$\int ydy=\int (-x+c)dx$
$=> \large\frac{y^2}{2}=\large\frac{-x^2}{2}$$+cx$
Rearranging the terms we get
$\large\frac{x^2}{2}+\frac{y^2}{2}$$-cx=0$
$=>x^2+y^2-2cx=0\qquad [(x^2-2(x)=(x-c)^2-c^2]$
$(ie)(x-c)^2-c^2+y^2=0$
$=>(x-c)^2+y^2=c^2$
Clearly this is an equation of a circle with center$(c,0)$ and radius=$c$
Hence the correct option is $D$
answered May 16, 2013 by meena.p
 

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