I was wondering if I could solve the equation $\iiint y(x)dxdxdx +\iint y(x)dxdx + \int y(x)dx =0$ as a differential equation, so I derivated it 3 times and got to $y''+ y'+ y = 0$.
The solution of the ode is $y(x)=C_1 e^{\omega x}+C_2 e^{\bar\omega x}$ where $\omega=\dfrac{-1+i\sqrt3}2$.
substituting in the original equation I got to
$C_1 e^{\omega x}\left(1+\dfrac{1}{\omega}+\dfrac{1}{\omega^2}\right)+C_2 e^{\bar\omega x}\left(1+\dfrac{1}{\bar\omega}+\dfrac{1}{\bar\omega^2}\right)+C_3(1+x)+C_4=0$
Is it legit, or do I have to use a fractional integral to solve the original equation?
P.S. Is the solution even possible without integration limits?