# Math Help - Find a differential equation given solution

1. ## Find a differential equation given solution

Find a di®erential equation whose solution is the two parameter fam-
ily of curves ln
y = c1*x^2+ c2.

I've gotten this far:
y= e^(c1*x^2+ c2)

y'= 2*c1*x* (e^(c1*x^2+ c2).

I also found y''. But for the world of it I cannot figure out the differential equation from y, y' and y''.

2. $\ln y=C_1 x^2+C_2\qquad \bigg/ \frac{d}{dx}$
$\frac{1}{y}y'=2C_1 x\qquad \bigg/ \frac{d}{dx}$
$\frac{yy''-y'^{{}^2}}{y^2}=2C_1 \bigg/ \cdot y^2$
$yy''-y'^{{}^2}=2C_1y^2$ or if you prefer $yy''-y'^{{}^2}-Cy^2=0.$