# Transforms of Discontinuous and Periodic Functions

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.6 – Transforms of Discontinuous and Periodic Functions

## Expected Educational Results

• Objective 23–1: I can sketch step and window functions.

• Objective 23–2: I can evaluate the Laplace transform of step and window functions.

• Objective 23–3: I can sketch periodic functions.

• Objective 23–4: I can evaluate the Laplace transform of periodic functions.

• Objective 23–5: I can evaluate the Laplace transform of the Gamma function.

## Transforms of Discontinuous and Periodic Functions

### Solving IVPs with Discontinuous Inputs

#### Investigation 13

Solve the following:

1. $\begin{array}{c}{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)+2y\left(t\right)=\left\{\begin{array}{ll}\left(t-1{\right)}^{2}& 1\le t<3\\ 0& \text{elsewhere}\end{array}\end{array}$$\displaystyle y^{\,\prime}(t)+2y(t)= \begin{cases} (t-1)^2 & 1\leq t < 3 \\ 0 & \text{elsewhere} \end{cases}$, $y\left(0\right)=0$$y(0)=0$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=1$$y^{\,\prime}(0)=1$

2. $\begin{array}{c}{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)+2y\left(t\right)=\left\{\begin{array}{ll}0& 0\le t<1\\ t-1& 1\le t<2\\ 1& t\ge 2\end{array}\end{array}$$\displaystyle y^{\,\prime}(t)+2y(t)= \begin{cases} 0 & 0\leq t < 1 \\ t-1 & 1\leq t < 2 \\ 1 & t\geq 2 \end{cases}$, $y\left(0\right)=0$$y(0)=0$

3. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+4y\left(t\right)=u\left(t-1\right)\mathrm{cos}\left(t\right)$$\displaystyle y^{\,\prime\prime}(t)+4y(t)=u(t-1)\cos{(t)}$, $y\left(0\right)=1$$y(0)=1$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=2$$y^{\,\prime}(0)=2$

4. $\begin{array}{c}2{y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+{y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)+2y\left(t\right)=\left\{\begin{array}{ll}1& 5\le t\le 20\\ 0& \text{elsewhere}\end{array}\end{array}$$\displaystyle 2y^{\,\prime\prime}(t)+y^{\,\prime}(t)+2y(t)= \begin{cases} 1 & 5\leq t \leq 20 \\ 0 & \text{elsewhere} \end{cases}$, $y\left(0\right)=0$$y(0)=0$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=0$$y^{\,\prime}(0)=0$

5. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+y\left(t\right)=10\left[u\left(t-\pi \right)-u\left(t-2\pi \right)\right]$$\displaystyle y^{\,\prime\prime}(t)+y(t)=10[u(t-\pi)-u(t-2\pi)]$, $y\left(0\right)=$$y(0)=$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=1$$y^{\,\prime}(0)=1$

6. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+y\left(t\right)=f\left(t\right)$$\displaystyle y^{\,\prime\prime}(t)+y(t)=f(t)$, $y\left(0\right)=0$$y(0)=0$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=0$$y^{\,\prime}(0)=0$, where $f\left(t\right)=t$$f(t)=t$ on $0$0 with period 2.

7. ${y}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(t\right)+2{y}^{\phantom{\rule{0.167em}{0ex}}\prime }+3y\left(t\right)=f\left(t\right)$$\displaystyle y^{\,\prime\prime}(t)+2y^{\,\prime}+3y(t)=f(t)$, $y\left(0\right)=0$$y(0)=0$, ${y}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(0\right)=0$$y^{\,\prime}(0)=0$, where $f\left(t\right)={e}^{2t}$$f(t)=e^{2t}$ on $0$0 with period 1.