#______x(t)=2cos t - 3sin t x(0)=7/2\nto advance in the circuit, locate x(π/2).

Explanation:

Step1: Integrate (x'(t))

$$ \begin{align*} x(t)&=\int(2\cos t - 3\sin t)dt\ &=2\int\cos tdt-3\int\sin tdt\ &=2\sin t + 3\cos t + C \end{align*} $$

Step2: Use the initial condition (x(0)=\frac{7}{2})

Substitute (t = 0) and (x(0)=\frac{7}{2}) into (x(t)=2\sin t + 3\cos t + C) $$ \begin{align*} \frac{7}{2}&=2\sin(0)+3\cos(0)+C\ \frac{7}{2}&=0 + 3\times1+C\ C&=\frac{7}{2}- 3=\frac{1}{2} \end{align*} $$ So (x(t)=2\sin t + 3\cos t+\frac{1}{2})

Step3: Find (x(\frac{\pi}{2}))

Substitute (t=\frac{\pi}{2}) into (x(t)) $$ \begin{align*} x\left(\frac{\pi}{2}\right)&=2\sin\left(\frac{\pi}{2}\right)+3\cos\left(\frac{\pi}{2}\right)+\frac{1}{2}\ &=2\times1+3\times0+\frac{1}{2}\ &=2+\frac{1}{2}=\frac{4 + 1}{2}=\frac{5}{2} \end{align*} $$

Answer:

(\frac{5}{2})