Answer to Question #217624 in Differential Equations for Fhumu

Question

d^2q/dt^2+16dq/dt=60
In an L-C circuit, L = 1 henry, C = 1
16 farad and E(t) = 60 volt.
Solce using laplace

dt dt represents the capacitor charge at anytime t.
Given q(0)= 0and q' ( 0)=i(0)=0 find:
7.1 the charge q on the capacitor at anytime t. (7)
7.2 the current i at anytime t.

Answer 1

$$s^2Y(s)+16sY(s)=\dfrac{60}{s}$$

$$Y(s)=\dfrac{60}{s^2(s+16)}$$

$$\dfrac{60}{s^2(s+16)}=\dfrac{A}{s+16}+\dfrac{B}{s}+\dfrac{C}{s^2}$$

$$=\dfrac{As^2+Bs(s+16)+C(s+16)}{s^2(s+16)}$$

$$s=0:C=\dfrac{15}{4}$$

$$s=-16:A=\dfrac{15}{64}$$

$$B=-A=-\dfrac{15}{64}$$

7.1

$$q(t)=\dfrac{15}{4}t-\dfrac{15}{64}+\dfrac{15e^{-16t}}{64}$$

7.2

$$i(t)=\dfrac{15}{4}-\dfrac{15e^{-16t}}{4}$$