I've a series circuit of a coil and a capacitor, in between those components we've a switch that will close when $t=0$. We can write:
$$ \begin{cases} \text{U}_\text{C}\left(t\right)=-\text{U}_\text{L}\left(t\right)\\ \\ \text{I}_\text{C}\left(t\right)=\text{U}'_\text{C}\left(t\right)\cdot\text{C}\\ \\ \text{U}_\text{L}\left(t\right)=\text{I}'_\text{L}\left(t\right)\cdot\text{L}\\ \\ \text{I}\left(t\right)=\text{I}_\text{C}\left(t\right)=\text{I}_\text{L}\left(t\right)\\ \end{cases}\space\space\space\space\space\therefore\space\space\space\space\space\frac{1}{\text{C}}\cdot\text{I}\left(t\right)=-\text{L}\cdot\text{I}''\left(t\right)\tag1 $$
Using Laplace transform:
- $$\text{I}\left(\text{s}\right)=\frac{\text{s}\cdot\text{I}\left(0\right)+\text{I}'\left(0\right)}{\frac{1}{\text{C}}+\text{L}\cdot\text{s}}\tag2$$
- $$\text{U}_\text{C}\left(\text{s}\right)=\frac{1}{\text{C}\cdot\text{s}}\cdot\left\{\frac{\text{s}\cdot\text{I}\left(0\right)+\text{I}'\left(0\right)}{\frac{1}{\text{C}}+\text{L}\cdot\text{s}}+\text{C}\cdot\text{U}_\text{C}\left(0\right)\right\}\tag3$$
- $$\text{U}_\text{L}\left(\text{s}\right)=\text{s}\cdot\text{L}\cdot\frac{\text{s}\cdot\text{I}\left(0\right)+\text{I}'\left(0\right)}{\frac{1}{\text{C}}+\text{L}\cdot\text{s}}-\text{L}\cdot\text{I}\left(0\right)\tag4$$
Well, I know that:
- $$\text{U}_\text{C}\left(0\right)=200\tag5$$
- $$\pi\sqrt{\text{C}\cdot\text{L}}<10\cdot10^{-6}=10^{-5}\space\Longleftrightarrow\space\text{C}\cdot\text{L}<\frac{10^{-10}}{\pi^2}\tag6$$
Question: How can I find the value of $\text{C}$ and $\text{L}$ using the things I know?