Question

The differential equation of all parabolas with axis parallel to the axis of $y$ is:

• A. $y_2=2y_1$
• B. $y_3=2y_1$
• C. $y_2^3=y_1$
• D. none of these

Answer 1

The correct option is D none of these

Let $(u,v)$ be the vertex of the parabola.
The equation of the required parabolas are of the form
${(x-u)}^2=A(y-v)$.
There are three unknown constants.
Differentiating successively three times we have
$2(x-u)=A{y}_1;2=A{y}_2$ and $y_3=0$.