Consider a circuit with $L_{1}$ and $L_{2}$ as inductors and $C_{1}$ and $C_{2}$ as the capacitors.
$I$ and $V$ are the manifest variables. Then write $I_{L_{1}}, I_{L_{2}}, I_{C_{1}}, I_{C_{2}}, V_{L_{1}}, V_{L_{2}}, V_{C_{1}}, V_{C_{2}}$ as the latent variables.
Using Kirchoff's current and voltage laws, I deduce
\begin{equation}\begin{cases}I=I_{L_{1}}+I_{L_{2}}\\ I_{L_{1}}=I_{C_{1}}\\ I_{L_{2}}=I_{C_{2}}\\ I_{C_{1}}+I_{C_{2}}=I\end{cases}\end{equation}
\begin{equation}\begin{cases}V=V_{L_{1}}+V_{C_{1}}\\ V=V_{L_{2}}+V_{C_{2}}\\ V_{L_{1}}+V_{C_{1}}=V_{L_{2}}+V_{C_{2}}\end{cases}\end{equation}
\begin{equation}\begin{cases}L_{1}\frac{dI_{L_{1}}}{dt}=V_{L_{1}}\\ L_{2}\frac{dI_{L_{2}}}{dt}=V_{L_{2}}\\ C_{1}\frac{dV_{C_{1}}}{dt}=I_{C_{1}}\\ C_{2}\frac{dV_{C_{2}}}{dt}=I_{C_{2}}\end{cases}\end{equation}
Then, after some elimination, I end up with
\begin{equation}\begin{cases} I=I_{L_{1}}+I_{L_{2}} \\ I_{L_{1}}=C_{1}\frac{dV_{C_{1}}}{dt} \\ I_{L_{2}}=C_{2}\frac{dV_{C_{2}}}{dt}\end{cases} \end{equation}
And \begin{equation} \begin{cases} V={L_{1}}\frac{dI_{L_{1}}}{dt}+V_{C_{1}} \\ V=L_{2}\frac{dI_{L_{2}}}{dt}+V_{C_{2}} \end{cases} \end{equation}
But now I'm stuck, because apart from substituting for $I_{L_{1}}$ and $I_{L_{2}}$, I can't see how to go any further. I want a single differential equation without the latent variables.
This is commonly attacked via the Laplace transform. In this formulation, the impedance of the elements are $Z_{L}= s L$ and $Z_{L}= 1/s C$, hence the total impendance (paralell of two series) is
$$\frac{V}{I}=Z = \left(\frac{1}{s L_1+ \frac{1}{sC_1}}+\frac{1}{s L_2+ \frac{1}{sC_2}}\right)^{-1} $$ Or
$$I = V \left(\frac{s \, C_1}{s^2 L_1 C_1+ 1}+\frac{s \, C_2}{s^2 L_2 C_2+ 1}\right)$$
From this you can (if you really want) obtain the differential equation that links $V_t$ with $I_t$. Also, you can evaluate the impedance $Z$ at specific values of the frequency, as a complex number ($s \to j \omega$) (the real part would correspond to the resistive part; the modulus would give you the ratio of peak or rms values of $V_t$ and $I_t$, assuming they are sinusoids, and so on).