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}{\displaystyle y^{\prime }(t)+2y(t)=\{\begin{array}{ll}(t-1{)}^2& 1\le t<3\\ 0& \text{elsewhere}\end{array}}\end{array}$$\displaystyle y^{\,\prime}(t)+2y(t)= \begin{cases} (t-1)^2 & 1\leq t < 3 \\ 0 & \text{elsewhere} \end{cases}$, $y(0)=0$$y(0)=0$, $y^{\prime }(0)=1$$y^{\,\prime}(0)=1$

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

3. ${\displaystyle y^{\prime \prime }(t)+4y(t)=u(t-1)\cos (t)}$$\displaystyle y^{\,\prime\prime}(t)+4y(t)=u(t-1)\cos{(t)}$, $y(0)=1$$y(0)=1$, $y^{\prime }(0)=2$$y^{\,\prime}(0)=2$

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

5. ${\displaystyle y^{\prime \prime }(t)+y(t)=10[u(t-\pi )-u(t-2\pi )]}$$\displaystyle y^{\,\prime\prime}(t)+y(t)=10[u(t-\pi)-u(t-2\pi)]$, $y(0)=$$y(0)=$, $y^{\prime }(0)=1$$y^{\,\prime}(0)=1$

6. ${\displaystyle y^{\prime \prime }(t)+y(t)=f(t)}$$\displaystyle y^{\,\prime\prime}(t)+y(t)=f(t)$, $y(0)=0$$y(0)=0$, $y^{\prime }(0)=0$$y^{\,\prime}(0)=0$, where $f(t)=t$$f(t)=t$ on $0<t<2$$0<t<2$ with period 2.

7. ${\displaystyle y^{\prime \prime }(t)+2y^{\prime }+3y(t)=f(t)}$$\displaystyle y^{\,\prime\prime}(t)+2y^{\,\prime}+3y(t)=f(t)$, $y(0)=0$$y(0)=0$, $y^{\prime }(0)=0$$y^{\,\prime}(0)=0$, where $f(t)=e^{2t}$$f(t)=e^{2t}$ on $0<t<1$$0<t<1$ with period 1.

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Last Modified: Sunday, 8 November 2020 22:02 EDT