Finding Time Constant without voltage source in diagram
The time constant \$\tau_L \$ can be found with \$\tau_L = \frac{L}{R_{eq}} \$ where \$R_{eq} \$ is the resistance found by taking out the inductor and looking at the equivalent resistance from the terminals.
The almost same argument goes for \$\tau_C \$, but it is found with \$\tau_C=R_{eq}C \$.