answer: \\frac{8}{e^{8}}\\#\\frac{da}{dt}=t + 4;a(2)=11.\\nparticular solution: \\nto advance in the…

answer: \\frac{8}{e^{8}}\\#\\frac{da}{dt}=t + 4;a(2)=11.\\nparticular solution: \\nto advance in the circuit, solve a(t)=-5.

answer: \\frac{8}{e^{8}}\\#\\frac{da}{dt}=t + 4;a(2)=11.\\nparticular solution: \\nto advance in the circuit, solve a(t)=-5.

Answer

Explanation:

Step1: Integrate the differential equation

We know that if (\frac{dA}{dt}=t + 4), then by the power rule of integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq- 1)), we have (A(t)=\int(t + 4)dt). [ \begin{align*} A(t)&=\int tdt+\int4dt\ &=\frac{t^{2}}{2}+4t + C \end{align*} ]

Step2: Use the initial condition (A(2) = 11) to find (C)

Substitute (t = 2) and (A(2)=11) into (A(t)=\frac{t^{2}}{2}+4t + C). [ \begin{align*} 11&=\frac{2^{2}}{2}+4\times2+C\ 11&=2 + 8+C\ 11&=10 + C\ C&=1 \end{align*} ] So (A(t)=\frac{t^{2}}{2}+4t + 1)

Step3: Solve (A(t)=-5)

Set (\frac{t^{2}}{2}+4t + 1=-5). Multiply through by (2) to get (t^{2}+8t+2=-10), or (t^{2}+8t + 12=0). Factor the quadratic equation: (t^{2}+8t + 12=(t + 2)(t+6)=0)

Answer:

(t=-2) or (t=-6)