When considering a second order differential equation, say: $$\frac{d^2y}{dx^2} = 10$$
is it possible to separate and integrate such? Separating the variables results in:
$$d^2y = 10 dx^2$$
and integrating results in:
$$y dy = 10xdx$$and then integrating once again results in:
$$\frac{y^2}{2} = 5x^2$$
Clearly solving for $y$ would yield something which is not the correct answer. I considered the scenario where the integral of $d^2y$ would be $dy$, but if that is the case why is the integral of $10dx^2$ = $10xdx$?
[Math] Correct Method for Second Order Separable Differential Equations
ordinary differential equations
Best Answer
$$\frac{d^2y}{dx^2} = 10$$ $d^2y = 10 dx^2$ is non-sens because $\frac{d^2y}{dx^2}$ is not a fraction but is a conventional symbol meaning that the function $y(x)$ is differentiated two times successively.
A more comprehensive writing is : $$\frac{d}{dx}\left(\frac{dy}{dx}\right)=10$$ Then you can separate : $$d\left(\frac{dy}{dx}\right)=10dx$$ Which is integrated as : $$\frac{dy}{dx}=10x+c_1$$ $$dy=(10x+c_1)dx$$ And integrated again : $$y=5x^2+c_1x+c_2$$