Solve the differential equations for the charge q(t) using Matlab. Choose carefu

Question

Solve the differential equations for the charge q(t) using Matlab. Choose carefully the interval of definition of the solution so that the behavior near t=0 and the convergence to the solution of the non-homogeneous are clear.

Example I. Suppose a resistor of 3 ohms, a capacitor of 5 × 10-3 farads, and an inductor of 10-2 henrys are connected in an electric circuit with a 110 volt, 60 Hz alternating current generator. Find the steady periodic current isp(t) Solution. We have R-3, C = 5 × 10-3, and L-10-2. Frequency 60 Hz means w (27) 60-120T, so our initial-value problem for q(t) is 10-2 q', + 3q, + 200 q = 110cos(120nt), q(0) =0=o(0).

Explanation / Answer

The matlab code to solve the differential equation is given below. This is solved using the dsolve function in matlab

syms q(t);
Dq=diff(q);
ode=0.01*diff(q,t,2)+3*diff(q,t)+200*q==110*cos(120*pi*t);
cond1=q(0)==0;
cond2=Dq(0)==0;
ysol(t)=dsolve(ode)

The solution is ysol(t) =

C1*exp(-100*t) + C2*exp(-200*t) + cos(atan(((13200*pi)/(14400*pi^2 + 10000) - (13200*pi)/(14400*pi^2 + 40000))/(11000/(14400*pi^2 + 10000) - 22000/(14400*pi^2 + 40000))) - 120*pi*t)*(((13200*pi)/(14400*pi^2 + 10000) - (13200*pi)/(14400*pi^2 + 40000))^2/(11000/(14400*pi^2 + 10000) - 22000/(14400*pi^2 + 40000))^2 + 1)^(1/2)*(11000/(14400*pi^2 + 10000) - 22000/(14400*pi^2 + 40000))

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