*5.2. The Exact Solution*

$$\begin{aligned} \text{Since } \mathbf{x}(0) &= 0 \text{, we obtain } \mathbf{x}(t) = I\_t^n(\iota III(t) - \iota III(t - T)) = \mathbf{x}^+(t) + \mathbf{x}^-(t). \\ \text{So } \mathbf{x}^+(t) &= \begin{array}{c} \mathbf{h}\_n(t) \* \iota III(t) \\ 0 \end{array} = \begin{array}{c} \int \frac{(t - \pi)^{n - 1}}{\Gamma(n)} l I H(\tau) d\tau \\ 0 \end{array} = \begin{array}{c} \frac{t^n}{\Gamma(n + 1)} l I H(t) \end{array} \text{ and } \\ \mathbf{x}^-(t) &= -h\_n(t) \* l I H(t - T) = -\frac{(t - T)^n}{\Gamma(n + 1)} l I H(t - T). \\ \text{Correspondently, see Figure 2 (for } n = 0.5): \end{aligned}$$

$$\mathbf{x}(t) = \begin{cases} \frac{t^n \mathbf{U}}{\Gamma(n+1)} \text{ for } 0 \le t \le T\\ \frac{\mathbf{U}}{\Gamma(n+1)} \left[ t^n - (t - T)^n \right] \text{ for } t \ge T \end{cases} \tag{35}$$

where *x*(*t*) *f or t* ≥ *T* represents the free response of (33) at *t*<sup>0</sup> = *T*.
