Chapter 5.7, Problem 19E

[M] Find formulas for the voltages v₁ and v₂ (as functions of time t) for the circuit in Example 1, assuming that R₁ = 1/5 ohm. R₂ = 1/3 ohm, C₁ = 4 farads, C₂ = 3 farads, and the initial charge on each capacitor is 4 volts.

EXAMPLE 1 The circuit in Figure 1 can be described by the differential equation

$\left[\begin{array}{c}{x^{\prime }}_1(t)\\ {x^{\prime }}_2(t)\end{array}\right]=\left[\begin{array}{cc}-(1/R_1+1/R_2)/C_1& 1/(R_2{C}_1)\\ 1/(R_2{C}_2)& -1/(R_2{C}_2)\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{x}_1(t)\\ x2(t)\end{array}\end{array}\right]$

where x₁(t) and x₂(t) are the voltages across the two capacitors at time t. Suppose resistor R₁. is 1 ohm, R₂ is 2 ohms, capacitor C₁ is 1 farad, and C₂ is .5 farad, and suppose there is an initial charge of 5 volts on capacitor C₁ and 4 volts on capacitor C₂. Find formulas for x₁(t) and x₂(t) that describe how the voltages change overtime.