TITLE: Is it valid to integrate on both sides in order to simplify a differential equation?
QUESTION [5 upvotes]: So I am attempting to solve this equation
$$y^{\prime\prime\prime} + y^{\prime\prime} +y^{\prime} = x+1. $$
Is it vaild to take the indefinite integral on both sides in order to simplify the expression and make it easier to solve?
\begin{align*}
\int (y^{\prime\prime\prime} + y^{\prime\prime} +y^{\prime})\, dx&= \int (x+1)\, dx\\
y^{\prime\prime} +y^{\prime} + y &= \frac{x^2}{2} +x + c
\end{align*}
Thanks!

REPLY [4 votes]: You can, but for the purpose of solving your specific equation, you may want to reserve that for the last step. In other words, let $z = y'$ then $$z''+z'+z=x+1$$
which is exact and does not have any constants floating around etc. It is also simpler since RHS is linear, not quadratic.
Now solve the equation and once you know what $z = z(x)$ is, you can find
$$
y(x) = \int y'(x) dx = \int z(x) dx,
$$
which will have the constant on the RHS...