Solution.
Due to Kirchhoff's low:
UL+UR+UC=E (1)
where
UL=Ldtdi=Ldt2d2q ; UR=iR=dtdqR ; UC=C1q .
Ldt2d2q+Rdtdq+C1q=E
dt2d2q+LRdtdq+LC1q=LE
LR=0.0520=400; LC1=0.05⋅100⋅10−61=2⋅105; LE=0.05100=2000
dt2d2q+400dtdq+2⋅105q=2000 (2)
dtdq= dtd(0.01−e−200t[0.01cos(400t)+0.02sin(400t)]) =e−200t(−6cos(400t)+8sin(400t));
dt2d2q=400e−200t(11cos(400t)+2sin(400t)).
Substitution all these into (2) gives:
400e−200t(11cos(400t)+2sin(400t))+ 400⋅e−200t(−6cos(400t)+8sin(400t))+2⋅105⋅ (0.01−e−200t[0.01cos(400t)+0.02sin(400t)])= 2000
We see that the left side is equal to the right one, which means q(t) satisfies Kirchhoff's law.
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