**Toolbox:**

- If we have an equation $f(x,y,c_1,c_2,....c_n)=u$ Containing n arbitrary constant $c_1,c_2...c_n$, then by differentiating n times, we get $(n+1)$ equations in total. If we eliminate the arbitrary constants $c_1,c_2....c_n,$ we get a D.E of order n

Step 1:

$y=ax^2 +bx+c$-----(i)

Where a,b are arbitrary constants

$\large\frac{dy}{dx}$$=2ax+b $ -----(ii)

$\large\frac{d^2y}{dx^2}$$=2a$-----(iii)

Step 2:

from (ii) and (iii)

$\qquad= \large\frac{1}{2} \frac{d^2 y}{dx^2}$

and $\large\frac{dy}{dx}=\frac{d^2 y}{dx^2}$$.x+b$

=>$b= \large\frac{dy}{dx}$$- x \large\frac{d^2 y}{dx^2}$

Step 3:

Substitute in (i) for a,b

$y= \large\frac{x^2}{2} \frac{d^2y}{dx^2}$$+x\bigg(\large\frac{dy}{dx}-x \frac{d^2y}{dx^2}\bigg)$$+c$

$\therefore \large\frac{d^2y}{dx^2}\bigg[ \frac{x^2}{2} $$-x^2\bigg]+x \large\frac{dy}{dx}$$-y +c=0$

$\therefore \large\frac{x^2}{2} \frac{d^2 y}{dx^2}-x \frac{dy}{dx}$$+y-c=0$ is the required D.E