Due to Kirchhoff's Law:
UR+UC=E (1)
where UR=iR=dtdqR , UC=Cq , E=100V.
Substitution UR and UC into (1) gives:
dtdqR+Cq=E
dtdq+RCq=RE
RC1=200⋅10−41=50Ω−1F−1
RE=200100=0.5ΩV
dtdq+50q=0.5 (2)
The solutions to a nonhomogeneous equation are of the form
q(t)=qh(t)+qp(t) ,
where qh is the general solution to the associated homogeneous equation and qp is a particular solution.
The associated homogeneous equation:
dtdq+50q=0
The general solution of this equation is determined by the roots of the characteristic equation:
λ+50=0
λ=−50
qh(t)=Ae−50t .
The particular solution of the differential equation:
qp(t)=B
qp′(t)=0
Substituting these into (2) we get:
0+50B=0.5
B=0.01.
q(t)=Ae−50t+0.01
Now we can apply initial condition to find A:
q(0)=0
q(0)=A+0.01=0
A=−0.01
q(t)=0.01(1−e−50t) ;
i(t)=q′(t)=0.5e−50t .
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