Laplace Transforms

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$, solve the given initial value problem using the method of Laplace transforms.

$\mathbf{1}·\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{2}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{4}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$, solve the given initial value problem using the method of Laplace transforms.

$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}$

In Problems $1-14$ , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{3}\mathbf{.}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{6}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{12}{\mathbf{e}}^{\mathbf{t}}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{7}$

In Problems 1-14 , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{w}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{w}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{;} \mathbf{w}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,} \mathbf{w}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$, solve the given initial value problem using the method of Laplace transforms

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{=}\mathbf{9}\mathbf{cost}\mathbf{+}\mathbf{7}\mathbf{sint}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{5}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{4}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$, solve the given initial value problem using the method of Laplace transforms

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{t}\mathbf{+}\mathbf{10}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{3}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms. 

$\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{z}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{z}\mathbf{=}\mathbf{21}{\mathbf{e}}^{\mathbf{t}\mathbf{-}\mathbf{1}}\mathbf{,} \mathbf{z}\left(\mathbf{1}\right)\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,} \mathbf{z}\mathbf{\text{'}}\left(\mathbf{1}\right)\mathbf{=}\mathbf{9}$

In Problems $1-14$, solve the given initial value problem using the method of Laplace transforms

$\mathbf{ }\mathbf{10}\mathbf{.}\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{t}\mathbf{-}\mathbf{8}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{;} \mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{5}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{;}\mathbf{y}\left(\mathbf{2}\right)\mathbf{=}\mathbf{3}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{2}\right)\mathbf{=}\mathbf{0}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms

$\mathbf{w}\mathbf{\text{'}}\mathbf{\text{'}}-\mathbf{2}\mathbf{w}\mathbf{\text{'}}\mathbf{+}\mathbf{w}\mathbf{=}\mathbf{6}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{;}\mathbf{w}\left(\mathbf{-}\mathbf{1}\right)\mathbf{=}\mathbf{3}\mathbf{;} \mathbf{w}\mathbf{\text{'}}\left(\mathbf{-}\mathbf{1}\right)\mathbf{=}\mathbf{7}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{8}\mathbf{cost}\mathbf{-}\mathbf{2}\mathbf{sint}\mathbf{;} \mathbf{y}\left(\frac{\mathbf{\pi }}{\mathbf{2}}\right)\mathbf{=}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\frac{\mathbf{\pi }}{\mathbf{2}}\right)\mathbf{=}\mathbf{0}$

 In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$, solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{;} \mathbf{y}\left(\mathbf{\pi }\right)\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{\pi }\right)\mathbf{=}\mathbf{0}$

In Problems $\mathbf{15}-\mathbf{24}$, solve for $\mathbf{Y}\left(\mathbf{s}\right)$ , the Laplace transform of the solution  $\mathbf{y}\left(\mathbf{t}\right)$ to the given initial value problem.

$\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{cost}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}$

Question: In Problems $\mathbf{15}-\mathbf{24}$, solve for $\mathbf{Y}\left(\mathbf{s}\right)$, the Laplace transform of the solution $\mathbf{y}\left(\mathbf{t}\right)$  to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems $\mathbf{15}-\mathbf{24}$, solve for $Y\left(s\right)$, the Laplace transform of the solution $y\left(t\right)$  to the given initial value problem.

$17.y^{\text{'}\text{'}}+y^{\text{'}}-y=t^3; y\left(0\right)=1, y^{\text{'}}\left(0\right)=0$

Determine the inverse Laplace transform  of the given function.

$\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{s}}\mathbf{(}\mathbf{4}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{)}}{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{)}}$

In Problems $\mathbf{15}-\mathbf{24}$ , solve for $\mathbf{Y}\left(\mathbf{s}\right)$ , the Laplace transform of the solution $y\left(t\right)$  to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}{\mathbf{e}}^{\mathbf{t}}\mathbf{;} \mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{3}$

In Problems $\mathbf{\text{15}}\text{-}\mathbf{\text{24}}$ , solve for , the Laplace transform of the solution  $\mathbf{\text{y}}\left(\mathbf{t}\right)$ to the given initial value problem.

In Problems $\mathbf{15}-\mathbf{24}$, solve for $\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$, the Laplace transform of the solution  $\mathbf{y}\left(\mathbf{t}\right)$ to the given initial value problem.

$y\text{'}\text{'}-2y\text{'}+y=cost-sint; y\left(\mathbf{0}\right)=1, y\text{'}\left(\mathbf{0}\right)=3$

In Problems \(21 - 30\), determine \({\mathcal{L}^{ - 1}}\left\{ F \right\}\).

\(F\left( s \right) = \frac{{6{s^2} - 13s + 2}}{{s\left( {s - 1} \right)\left( {s - 6} \right)}}\)

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms. $$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathit{\pi }\mathbf{)}\mathbf{-}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{2}\mathit{\pi }\mathbf{)} \\ \mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0} \end{gathered}$$

In Problems, determine ${\mathcal{L}}^{-1}\left\{F\right\}$ $s^{2}F\left(s\right)-4F\left(s\right)=\frac{5}{s+1}$

In Problems 21-30, determine ${\mathcal{L}}^{-1}\left\{F\right\}$

$s^{2}F\left(s\right)+sF\left(s\right)-6F\left(s\right)=\frac{s^2+4}{s^2+s}$

In Problems 21-30, determine ${\mathcal{L}}^{-1}\left\{F\right\}$

$sF\left(s\right)-F\left(s\right)=\frac{2s+5}{s^2+2s+1}$

In Problems 29 - 32, use the method of Laplace transforms to find a general solution to the given differential equation by   assuming $y\left(0\right)=a and y\text{'}\left(0\right)=b$ a and b are arbitrary constants.

$y\text{'}\text{'}-5y\text{'}+6y=-6t{e}^{2t}$

Determine the Laplace transform of each of the following functions:

$\left(a\right)f_1\left(t\right)=\left\{\begin{array}{l}t, t=1,2,3,....,\\ e^t, t\ne 1,2,3,....\end{array}\right.$

$\left(b\right)f_2\left(t\right)=\left\{\begin{array}{l}{e}^t, t\ne 5,8,\\ 6, t=5,\\ 0, t=8,\end{array}\right.$

$\left(c\right)f_3\left(t\right)=e^t.$

Which of the preceding functions is the inverse Laplace transform of $\frac{1}{\left(s-1\right)}?$

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as 

${\mathcal{L}}^{-1}\left\{\frac{d^{n}F}{d{s}^n}\right\}\left(t\right)={\left(-t\right)}^{n}f\left(t\right),$

where,$f={\mathcal{L}}^{-1}\left\{F\right\}$.Use this equation in Problems 33-36 to compute ${\mathcal{L}}^{-1}\left\{F\right\}.$$F\left(s\right)=ln\left(\frac{s+2}{s-5}\right)$

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as

${\mathcal{L}}^{-1}\left\{\frac{d^{n}F}{d{s}^n}\right\}\left(t\right)={\left(-t\right)}^{n}f\left(t\right),$

Where $f={\mathcal{L}}^{-1}\left\{F\right\}$.Use this equation in Problems 33-36 to compute ${\mathcal{L}}^{-1}\left\{F\right\}$.$F\left(s\right)=ln\left(\frac{s-4}{s-3}\right)$.

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as

${\mathcal{L}}^{-1}\left\{\frac{d^{n}F}{d{s}^n}\right\}\left(t\right)={\left(-t\right)}^{n}f\left(t\right)$,

Where  $f={\mathcal{L}}^{-1}\left(F\right)$.Use this equation in Problems 33-36 to compute ${\mathcal{L}}^{-1}\left(F\right).$$F\left(s\right)=ln\left(\frac{s^2+9}{s^2+1}\right)$

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as

${\mathcal{L}}^{-1}\left\{\frac{d^{n}F}{d{s}^n}\right\}\left(t\right)={\left(-t\right)}^{n}f\left(t\right)$

Where $f={\mathcal{L}}^{-1}\left\{F\right\}$.Use this equation in Problems 33-36 to compute

${\mathcal{L}}^{-1}\left\{F\right\}$. $F\left(s\right)=arctan\left(\frac{1}{s}\right)$

Prove Theorem 7, page 368, on the linearity of the inverse transform. [Hint: Show that the right-hand side of equation (3) is a continuous function on $[0,\infty )$  whose Laplace transform is  $\left[F_1\left(s\right)+F_2\left(s\right)\right]$

Residue Computation. Let $\frac{P\left(s\right)}{Q\left(s\right)}$   be a rational function with deg$P<degQ$   and suppose $s-r$  is a non-repeated linear factor of $Q\left(s\right)$ . Prove that the portion of the partial fraction expansion of $\frac{P\left(s\right)}{Q\left(s\right)}$ corresponding to $s-r$  is $\frac{A}{s-r}$, where $A$   (called the residue) is given by the formula

$A=\underset{s\to r}{lim}\frac{\left(s-r\right)P\left(s\right)}{Q\left(s\right)}=\frac{P\left(r\right)}{Q^{\text{'}}\left(r\right)}$

Use the residue computation formula derived in Problem 38 to determine quickly the partial fraction expansion for$F\left(s\right)=\frac{2s+1}{s\left(s-1\right)\left(s+2\right)}$

Heaviside's Expansion Formula. Let $P\left(s\right)$  and $Q\left(s\right)$  be polynomials with the degree of $P\left(s\right)$  less than the degree of $Q\left(s\right)$  . Let

$Q\left(s\right) =\left(s-r_1\right) \left(s-r_2\right)...\left(s-r_n\right)$ $, {\mathcal{L}}^{-1}\left\{\frac{P}{Q}\right\}\left(t\right)=\sum _{i=1}^n\frac{P\left(r_i\right)}{P\left(r_i\right)}{e}^{r_{i}t}$

Use the residue formulas derived in Problems 38 and 42 to determine the partial fraction expansion for

$F\left(s\right)=\frac{6s^2+28}{\left(s^2-2s+5\right)(s+2)}$

Use the Laplace transform table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {6{e^{ - 3t}} - {t^2} + 2t - 8} \right\}\)

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\[L\left\{ {{\bf{5 - }}{{\bf{e}}^{{\bf{2t}}}}{\bf{ + 6}}{{\bf{t}}^{\bf{2}}}} \right\}\]

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {{{\bf{t}}^{\bf{3}}}{\bf{ - t}}{{\bf{e}}^{\bf{t}}}{\bf{ + }}{{\bf{e}}^{{\bf{4t}}}}{\bf{cost}}} \right\}\)

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {{{\bf{t}}^{\bf{2}}}{\bf{ - 3t - 2}}{{\bf{e}}^{{\bf{ - t}}}}{\bf{sin3t}}} \right\}\)

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {{{\bf{e}}^{{\bf{3t}}}}{\bf{sin6t - }}{{\bf{t}}^{\bf{3}}}{\bf{ + }}{{\bf{e}}^{\bf{t}}}} \right\}\)

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {{{\bf{t}}^{\bf{4}}}{\bf{ - }}{{\bf{t}}^{\bf{2}}}{\bf{ - t + sin}}\sqrt {\bf{2}} {\bf{t}}} \right\}\)

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {{{\bf{t}}^{\bf{4}}}{{\bf{e}}^{{\bf{5t}}}}{\bf{ - }}{{\bf{e}}^{\bf{t}}}{\bf{cos}}\sqrt {\bf{7}} {\bf{t}}} \right\}\)

Use the Laplace transformation table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {{{\bf{e}}^{{\bf{ - 2t}}}}{\bf{cos}}\sqrt {\bf{3}} {\bf{t - }}{{\bf{t}}^{\bf{2}}}{{\bf{e}}^{{\bf{ - 2t}}}}} \right\}\)

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{t}}^{\mathbf{2}}\mathbf{+}{\mathbf{e}}^{\mathbf{t}}\mathbf{sin}\mathbf{2}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{3}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{cos}\mathbf{3}\mathbf{t}\mathbf{+}{\mathbf{e}}^{\mathbf{6}\mathbf{t}}\mathbf{-}\mathbf{1}$.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{3}{\mathbf{t}}^{\mathbf{4}}\mathbf{-}\mathbf{2}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{2}{\mathbf{t}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{-}\mathbf{t}\mathbf{+}\mathbf{cos}\mathbf{4}\mathbf{t}$.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{sin}\mathbf{2}\mathbf{t}\mathbf{+}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}{\mathbf{t}}^{\mathbf{2}}$