Laplace transforms: R series with RC parallel circuit

Problem Statement

Find the Voltage across the capacitor for t>=0:

Voltage across capacitor at t(0-)=0

v(t0)=0 Volts

v(t)=10 Volts

R1=20 Ohm

R2=30 Ohm

C=.1 Farad

Loop 1

$v(t)=R1(i1+i2)+R2(i1)$

$10=20(i1+i2)+30(i1)$_______________________________________equation (1)

Loop 2

$v(t)=R1(i1+i2)+{\dfrac {1}{C}}\int {i2dt}$

$10=20(i1+i2)+{\dfrac {1}{0.1}}\int {i2dt}$_______________________equation (2)

Solving equations (1) and (2) simultaneously

Laplace Transform S-domain

Inverse Laplace transform T-domain

Voltage on Capacitor