Absolute value in integrating factor

Question

So the problem is:

$(3+t) x' + x = sin(t)$, $x(0) = 0$

The integrating factor is $(3+t)$, but shouldn't it be $|3+t|$? As e raised to the the integral of $1/(3+t)$ is $|3+t|$ not $(3+t)$? I suspect it has something to do with the initial condition, but I'm not sure.

Thanks

Answer 1

You get for the homogeneous equation, after exponentiation, $$ |x|=C·|3+t| $$ The only variant of sign combinations that is continuously differentiable everywhere is $x=C·(3+t)$.

Of course, one could also immediately see that the left side is $$ \frac{d}{dt}((3+t)·x(t)). $$