sir...an errror.. in ur soln..(though its very obvious)
3rd step shud not hav "x" in denominator.
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0 \\\frac{b^2}{a^2}+\frac{y}{x}\frac{dy}{dx}=0 \\
Now differentiate once to get the differential equation as you will have gotten rid of both a and b at this stage...
sir...an errror.. in ur soln..(though its very obvious)
3rd step shud not hav "x" in denominator.