Question

Form the differential equation of family of parabolas x²=4ay where 'a' is a arbritary constant ​

Answer 1

Dear student
$$\begin{gathered} \mathrm{Given}{\mathrm{x}}^2=4\mathrm{ay} \\ \Rightarrow \mathrm{a}=\frac{{\mathrm{x}}^2}{4\mathrm{y}} \\ \mathrm{Now}\mathrm{differentiating}\mathrm{wrt}\mathrm{x} \\ \frac{4\mathrm{y}\left(2\mathrm{x}\right)-{\mathrm{x}}^{2}4{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}}{{\left(4\mathrm{y}\right)}^2}=0\left[\mathrm{quotient}\mathrm{rule}\right] \\ 8\mathrm{xy}=4{\mathrm{x}}^2\mathrm{y}\text{'}\left[\mathrm{as}\mathrm{y}\text{'}=\frac{\mathrm{dy}}{\mathrm{dx}}\right] \\ 2\mathrm{y}=\mathrm{xy}\text{'} \end{gathered}$$
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