if we assume the condition immediately after switch is closed (t=0+), *Capacitor voltage cannot jump. *Inductor current cannot jump. dv(0+)/dt=i_c(0+)/c di(0+)/dt=v_l(0+)/L which means we can find the initial condition of the post-zero system algebraically. However, it contradcits to the definition of differentiability since the pre-zero value and post-zero value of the system are different. (I_c(0+)=/=i_c(0-), v_l(0+)=/=v_l(0-)) I wonder how we can justify the math of the system. Can I think this as two diffrent systems (ODEs) with one condition shared?