I am attempting to grasp the basics of separation of variables for a second order separable differential equation, and am failing to do so:
Given the equation:
$$ x=\frac{d^2y}{dx^2}$$
I know from calculus class intuitively that the solution is:
$$ y=\frac{x^3}{6}$$
But if I am to use separation of variables, why don't I get:
$$ \frac{y^2}{2}=\frac{x^3}{6} $$
Since I should have had to integrate the $y$ side twice as well.
Best Answer
You don't have something like $f(x)dx dx=g(y)dy dy$. You have $f(x) dx =g(y)d\left(\frac{dy}{dx}\right)$. Remember that the second deriviative is $\frac{d \left(\frac{dy}{dx}\right)}{dx}$. The actual solution is $\frac{x^3}6+cx+k$, where $c$ and $k$ are constants. This should be your process:
$x=\frac{d^2y}{dx^2}$
$x dx=d\left(\frac{dy}{dx}\right)$
$\int xdx=\int d\left(\frac{dy}{dx}\right)$
$\frac{x^2}2=\frac{dy}{dx}$