given is the following first order ODE:
$\dot\epsilon(t) = \frac{1}{\eta}\cdot\sigma(t) + \frac{1}{E_1}\cdot\dot\sigma(t)$,
where $\eta$ and $E_1$ are constants. The initial conditions are: $\sigma(t = 0) = \sigma_0$; $\dot\sigma(t = 0) = 0$ and $\epsilon(t = 0) = \epsilon_0 = \frac{\sigma_0}{E_1}$ as well as $\dot\epsilon(t = 0) = 0$
The "function" $\sigma(t)$ is also given as $\sigma(t) = \sigma_0 = \text{const.}$
It's quite simple to solve this Equation by separation with the usage of the initial conditions. This gives:
$\epsilon(t) = \frac{\sigma_0}{E_1}+\frac{\sigma_0}{\eta}\cdot t$
Actually I want to solve the problem with Laplace-transformation. This leads me to (with $p_0 = \frac{1}{\eta}$ and $p_1 = \frac{1}{E_1}$):
$\mathcal{L}\{\epsilon\}(s)\cdot s-\epsilon_0 = p_0 \cdot \mathcal{L}\{\sigma\}(s) + p_1\cdot\bigl[\mathcal{L}\{\sigma\}(s)\cdot s - \sigma_0\bigr]$
... and now I don't know the way back; I tried it like the following:
$\int_0^t\epsilon(\xi)\delta(\xi)\text{d}\xi - \epsilon_0\delta(t) = p_0\cdot\int_0^t\sigma(\xi)\text{d}\xi + p_1\cdot\int_0^t\sigma(\xi)\delta(\xi)\text{d}\xi-p_1\cdot\sigma_0\delta(t)$
... but there's something wrong, 'cause this leads not to the same solution like given above.
Maybe someone could tell me my mistake. That would be nice!
Thanks in advance!
You have just one differential equation with two dependent variables $\epsilon(t)$ and $\sigma(t)$. That's not enough: you need another differential equation if you want to specify a solution uniquely.
EDIT: If you know $\sigma$ and want $\epsilon$, you get $$ {\mathcal L}\{\epsilon\}(s) = \left(\dfrac{1}{s\eta} + \dfrac{1}{E_1}\right) {\mathcal L}\{\sigma\}(s) + \dfrac{\epsilon_0 E_1 - \sigma_0}{E_1 s}$$ You said $\epsilon_0 = \sigma_0/E_1$ so the second term on the right is $0$.
You might note that the Laplace transform of $\displaystyle\int_0^t g(\xi) \; d\xi$ is $\dfrac{1}{s} {\mathcal L}\{g\}(s)$ so indeed $$ \epsilon(t) = \dfrac{1}{\eta} \int_0^t \sigma(\xi)\; d\xi + \dfrac{1}{E_1} \sigma(t)$$ which is what you would have obtained by integration without detouring through Laplace transforms.