I have a simple first order ODE with the form: $$ f'(t) + Pf(t) = Q(t) $$ I am trying to solve for $f(t)$. P is a constant, but it could be a function of t as well in more complex versions of the function, while Q(t) contains a sum of constants and another function of t.
So far I solved ODEs using either Laplace Transform + Partial Fractions with a lookup table or Integrating Factors to solve simple functions. However, the time frame of the function I am trying to derive occurs after $t \geq t_0$ and the initial condition is therefore given by $f(t_0)$.
Since I don't know $f(0)$, when I try to apply Laplace, I cannot simply transform $f'(t)$ to $sF(s) - f(0)$ and end up with an equation which would be trivial to transform back:
$$ sF(s) - f(0) + PF(s) = Q(s)$$ $$ F(s)(s+P) = Q(s) + f(0) $$
I tried to introduce a time shift by $t_0$, which would lead to a transformation like $e^{t_0s}[sF(s) - f(t_0)]$, but when I bring all factors together, my equation gets very messy since I then come to an expression of the form:
$$F(s) = \frac{Q(s)}{s e^{t_0s} + P} + f(t_0)\frac{e^{t_0s}}{{s e^{t_0s} + P}} $$
I haven't been able to solve the right side, so I feel that I am probably doing something wrong.
Is there a way to substitute these new boundaries into the rest of the equation to get a relatively straightforward solution like we could if $f(0)$ was given?
Would a completely different approach be better altogether?
The integrating factor method is very straightforward for boundary conditions not given at $t=0$. Just start your integration at $t_0$ instead: \begin{eqnarray} f'(t)+P f(t) = Q(t)&\Longrightarrow& \frac{d}{dt}\left[e^{Pt}f(t)\right] = Q(t)e^{Pt} \Longrightarrow e^{Pt}f(t) - e^{P t_0}f(t_0) = \int_{t_0}^t Q(\tau) e^{P\tau}d\tau \\&\Longrightarrow& f(t) = f(t_0)e^{P(t_0-t)} + \int_{t_0}^t Q(\tau)e^{P(\tau-t)}d\tau. \end{eqnarray} If $P$ is not constant, doing the above procedure results in the same expression, except the $P(t_2-t_1)$ terms in the exponentials are replaced with $\int_{t_1}^{t_2} P(T)dT$.
I think that the usual treatment of integrating factors using indefinite integrals obscures how the boundary conditions factor in. Indefinite integrals can only give a family of solutions. Any time you need a particular solution, those integrals are going to have to become definite.
I don't think the Laplace transform is well-suited to this, but if you must, I'd suggest letting $f(t) = g(t - t_0)$, so that $g$'s boundary condition actually is at $t = 0$, then transforming.