Differential equation mixed RLC-circuit, C parallel to RL
The voltage, \$ v\$, across \$\small C\$ is equal to the the voltage across the \$\small R, L\$ series combination.

The current in \$\small C\$ is \$i_1=\small C\large \frac{dv}{dt}\$

The current in \$\small RL\$ is related to \$v\$ by; \$ v=\small R\large i_2+\small L\large\frac{di_2}{dt}\$, where \$i_2\$ is the current through \$\small R\$ and \$\small L\$.

Solve both ODEs for \$ i_1\$ and \$ i_2\$, and the total current in the \$\small RLC\$ circuit is then \$i=i_1 + i_2\$.

Let \$\small t\rightarrow \infty\$ in the real parts of the exponentials to remove any transients, and you now have \$ i\$ and \$v\$ in the required steady-state sinusoidal forms.