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Find the differential equation of family of straight lines $y=mx+\large\frac{a}{m}$ When $m$ is the parameter.

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  • If we have an equation $f(x,y,c_1,c_2,....c_n)=u$ Containing n arbitrary constant $c_1,c_2...c_n$, then by differentiating n times, we get $(n+1)$ equations in total. If we eliminate the arbitrary constants $c_1,c_2....c_n,$ we get a D.E of order n
Step 1:
$y=mx+\large\frac{a}{m}$ where m is the parameter
$\large\frac{dy}{dx}$$=m $ -----(ii)
Step 2:
Substitute for m from (ii) in (i)
$\therefore y= x \large\frac{dy}{dx}+ \large\frac{a}{\Large\frac{dy}{dx}}$
(ie) $x \bigg( \large\frac{dy}{dx}\bigg)^2$$-y \large\frac{dy}{dx}$$+a =0$
It is the required D.E
answered Sep 3, 2013 by meena.p

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