7.6E: Convolution (Exercises)

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Q7.6.1

1. Express the inverse transform as an integral.

1. $$1\over s^2(s^2+4)$$
2. $$s\over(s+2)(s^2+9)$$
3. $$s\over(s^2+4)(s^2+9)$$
4. $$s\over(s^2+1)^2$$
5. $$1\over s(s-a)$$
6. $$1\over(s+1)(s^2+2s+2)$$
7. $$1\over (s+1)^2(s^2+4s+5)$$
8. $$1\over(s-1)^3(s+2)^2$$
9. $$s-1\over s^2(s^2-2s+2)$$
10. $$s(s+3)\over(s^2+4)(s^2+6s+10)$$
11. $$1\over(s-3)^5s^6$$
12. $$1\over(s-1)^3(s^2+4)$$
13. $$1\over s^2(s-2)^3$$
14. $$1\over s^7(s-2)^6$$

2. Find the Laplace transform.

1. $$\int_0^t\sin a\tau\cos b(t-\tau)\, d\tau$$
2. $$\int_0^t e^\tau\sin a(t-\tau)\,d\tau$$
3. $$\int_0^t\sinh a\tau\cosh a(t-\tau)\,d\tau$$
4. $$\int_0^t\tau(t-\tau)\sin \omega\tau\cos\omega (t-\tau)\,d\tau$$
5. $$e^t\int_0^t\sin\omega\tau \cos\omega (t-\tau)\,d\tau$$
6. $$e^t\int_0^t\tau^2 (t-\tau)e^\tau\,d\tau$$
7. $$e^{-t}\int_0^t e^{-\tau}\tau\cos\omega (t-\tau)\,d\tau$$
8. $$e^t\int_0^t e^{2\tau}\sinh (t-\tau)\,d\tau$$
9. $$\int_0^t\tau e^{2\tau}\sin 2(t-\tau)\,d\tau$$
10. $$\int_0^t (t-\tau)^3 e^\tau\, d\tau$$
11. $$\int_0^t\tau^6 e^{-(t-\tau)}\sin 3(t-\tau)\,d\tau$$
12. $$\int_0^t\tau^2 (t-\tau)^3\, d\tau$$
13. $$\int_0^t (t-\tau)^7 e^{-\tau} \sin 2\tau\,d\tau$$
14. $$\int_0^t (t-\tau)^4\sin 2\tau\,d\tau$$

3. Find a formula for the solution of the initial value problem.

1. $$y''+3y'+y=f(t),\quad y(0)=0,\quad y'(0)=0$$
2. $$y''+4y=f(t),\quad y(0)=0,\quad y'(0)=0$$
3. $$y''+2y'+y=f(t),\quad y(0)=0,\quad y'(0)=0$$
4. $$y''+k^2y=f(t),\quad y(0)=1,\quad y'(0)=-1$$
5. $$y''+6y'+9y=f(t),\quad y(0)=0,\quad y'(0)=-2$$
6. $$y''-4y=f(t),\quad y(0)=0,\quad y'(0)=3$$
7. $$y''-5y'+6y=f(t),\quad y(0)=1,\quad y'(0)=3$$
8. $$y''+\omega^2y=f(t),\quad y(0)=k_0,\quad y'(0)=k_1$$

4. Solve the integral equation.

1. $$y(t)=t-\int_0^t (t-\tau) y(\tau)\,d\tau$$
2. $$y(t)=\sin t-2 \int_0^t\cos (t-\tau) y (\tau)\,d\tau$$
3. $$y(t)=1+2 \int_0^ty(\tau)\cos(t-\tau)\,d\tau$$
4. $$y(t)=t+\int_0^t y(\tau)e^{-(t-\tau)}\,d\tau$$
5. $$y'(t)=t+\int_0^t y(\tau)\cos (t-\tau)\,d\tau,\, y(0)=4$$
6. $$y(t)=\cos t-\sin t+ \int_0^t y(\tau)\sin (t-\tau)\,d\tau$$

5. Use the convolution theorem to evaluate the integral.

1. $$\int_0^t (t-\tau)^7\tau^8\, d\tau$$
2. $$\int_0^t(t-\tau)^{13}\tau^7\,d\tau$$
3. $$\int_0^t(t-\tau)^6\tau^7\, d\tau$$
4. $$\int_0^te^{-\tau}\sin(t-\tau)\,d\tau$$
5. $$\int_0^t\sin\tau\cos2(t-\tau)\,d\tau$$

6. Show that

$\int_0^tf(t-\tau)g(\tau)\,d\tau=\int_0^tf(\tau)g(t-\tau)\,d\tau\nonumber$

by introducing the new variable of integration $$x=t-\tau$$ in the first integral.

7. Use the convolution theorem to show that if $$f(t)\leftrightarrow F(s)$$ then

$\int_0^tf(\tau)\,d\tau\leftrightarrow {F(s)\over s}.\nonumber$

8. Show that if $$p(s)=as^2+bs+c$$ has distinct real zeros $$r_1$$ and $$r_2$$ then the solution of

$ay''+by'+cy=f(t),\quad y(0)=k_0,\quad y'(0)=k_1\nonumber$

is

\begin{aligned} y(t)&=\; k_0{r_2e^{r_1t}-r_1e^{r_2t}\over r_2-r_1}+k_1{e^{r_2t}-e^{r_1t} \over r_2-r_1} \\ &+{1\over a(r_2-r_1)}\int_0^t(e^{r_2\tau}-e^{r_1\tau})f(t-\tau)\,d\tau.\end{aligned}\nonumber

9. Show that if $$p(s)=as^2+bs+c$$ has a repeated real zero $$r_1$$ then the solution of

$ay''+by'+cy=f(t),\quad y(0)=k_0,\quad y'(0)=k_1\nonumber$

is

$y(t)=\; k_0(1-r_1t)e^{r_1t}+k_1te^{r_1t} +{1\over a}\int_0^t\tau e^{r_1\tau}f(t-\tau)\,d\tau.\nonumber$

10. Show that if $$p(s)=as^2+bs+c$$ has complex conjugate zeros $$\lambda\pm i\omega$$ then the solution of

$ay''+by'+cy=f(t),\quad y(0)=k_0,\quad y'(0)=k_1\nonumber$

is

\begin{aligned} y(t)&=\; e^{\lambda t}\left[k_0(\cos\omega t-{\lambda\over\omega}\sin\omega t)+{k_1\over\omega}\sin\omega t\right] \\ &+{1\over a\omega}\int_0^te^{\lambda t}f(t-\tau)\sin\omega\tau\, d\tau.\end{aligned}\nonumber

This page titled 7.6E: Convolution (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.