I have trouble uderstanding a part of the solution of the ode below.
$y'(t)+y(t)=\int _{0} ^{2} y(t)dt$ (1)
The solution is this:
Let $u=\int _{0} ^{2} y(t)dt$
$(1) \Rightarrow y'(t)+y(t)=u$ (2)
The ode has intagrating factor of $μ(t)=e^t$
$\xrightarrow{(2)*μ(t)} e^ty'+e^ty=e^tu \Leftrightarrow(e^ty(t))'=e^tu $
$\Leftrightarrow e^ty(t)=\int e^tu dt$
Here's what I don't get, the answer goes on like this
$e^ty(t)=e^tu+c$
It seems to me that this would be the case if $u$ was not dependat to $t$ but since $u=\int _{0} ^{2} y(t)dt$ it's obviously dependant..
What am I missing here? Can someone explain this part of the solution?
$u=\int _{0} ^{2} y(t)dt$ is a number, not a function of $t$.