The following differential equations describe the behaviour of the current I(t) in a network following a...?

Question

switching action at time t=0.
In each case, how to I write the differential equation that must be solved to find the transient or natural circuit response?

dI(t)/dt + 6I(t) = 10

d^2I(t)/dt^2 + 5I(t) = 24

Many Thanks for your help :)

Answer 1

Are you asking how to solve these differential equations?
dI(t)/dt + 6I(t) = 10
Let J(t) = I(t)-5/3, we have:
dJ(t)/dt + 6J(t) = 0
The solution of this equation is: J(t) = C*exp(-6t)
or: I(t) = C*exp(-6t) + 5/3
where C is to be fixed with I(0) = C + 5/3

d^2I(t)/dt^2 + 5I(t) = 24
Let J(t) = I(t)-24/5, we have:
d^2J(t)/dt^2 + 5J(t) = 0
The solution of this equation is:
J(t) = C1*cos(sqrt(5)*t) + C2*sin(sqrt(5)*t)
or: I(t) = C1*cos(sqrt(5)*t) + C2*sin(sqrt(5)*t) + 24/5
where C's are to be fixed with I(0) = C1 + 24/5, and
dI(0)/dt = C2*sqrt(5)