Can anyone help me with this question, please?
Find a diŽerential equation whose solution is the family of parabolaswith vertices and foci on the x-axis.
Everything starts from the equation of the parabola. Parabolas with vertices and foci on the abscissa all have the following equation: $\displaystyle y^2=2px$ where $\displaystyle p$ is the distance between the focus and the origin of the coordinate system (in this case its the distance along the abscissa). Applying derivative with respect to $\displaystyle x$ and you get the following differential equation:$\displaystyle 2yy'=2p$ where you can substitute $\displaystyle 2p$ with general constant term $\displaystyle C$ providing the constraint $\displaystyle C>0$ since it carries the information about the distance which cannot be negative.
Hope this helps.