The differential equation of all parabolas whose axis is y-axis, is

Differential Equation of parabolas whose axis is \(\mathrm{Y}\) axis, is Let the vertex be \((0, \mathrm{k})\)


\(\begin{aligned}
& \therefore x^2=4 b y-4 b k \\
& \therefore 2 x=4 b \frac{d y}{d x}-0 \Rightarrow b=\left(\frac{x}{2}\right) \times \frac{1}{\left(\frac{d y}{d x}\right)} \\
& 2 x=4 b \frac{d y}{d x}-0 \Rightarrow b=\left(\frac{x}{2}\right) \times \frac{1}{\left(\frac{d y}{d x}\right)}
\end{aligned}\)
Substituting value of \(b\) in eq. (1), we get
\(\begin{aligned}
& \therefore \quad x^2=4\left(\frac{x}{2}\right) \times \frac{1}{\left(\frac{d y}{d x}\right)}(y-k) \\
& \therefore \quad \mathrm{x}^2\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=2 \mathrm{x}(\mathrm{y}-\mathrm{k}) \\
& \therefore \quad x^2\left(\frac{d^2 y}{d x^2}\right)+\left(\frac{d y}{d x}\right)(2 x)=2 x \frac{d y}{d x}+2(y-k) \\
& \therefore \quad \mathrm{y}-\mathrm{k}=\frac{\mathrm{x}^2\left(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}}\right)}{2} \\
&
\end{aligned}\)
Substituting value of \((\mathrm{y}-\mathrm{k})\) in eq. (1), we get
\(\begin{aligned}
& \left.x^2=4\left(\frac{x}{2}\right)\left[\frac{1}{\left(\frac{d y}{d x}\right)}\right]^{\left.\frac{x^2\left(\frac{d^2 y}{d^2}\right)}{2}\right]}\right] \\
& \therefore \quad x^2\left(\frac{d y}{d x}\right)=x^3\left(\frac{d^2 y}{d x^2}\right) \Rightarrow x\left(\frac{d^2 y}{d x^2}\right)-\left(\frac{d y}{d x}\right)=0
\end{aligned}\)