Chapter 7: The Laplace Transform

In Problems 15 and 16 use a graphing utility to graph the indicated solution.

$\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$of Problem 13 for$\mathbf{0}\le \mathbf{t}<\mathbf{3}\pi$.

In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function $\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{=}\mathbf{L}\mathbf{\left\{}\mathbf{y}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right\}}$. Solve the first-order DE for Y (S) and then find $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{L}}^{\mathbf{-}\mathbf{1}}\mathbf{\left\{}\mathbf{Y}\mathbf{\right(}\mathbf{s}\mathbf{\left)}\mathbf{\right\}}$.

$\mathbf{2}{\mathbf{y}}^{\text{'}\text{'}}\mathbf{+}{\mathbf{ty}}^{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{10}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{y}}^{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function $\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{=}\mathbf{L}\mathbf{\left\{}\mathbf{y}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right\}}$. Solve the first-order DE for Y (s) and then find $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{L}}^{\mathbf{-}\mathbf{1}}\mathbf{\left\{}\mathbf{Y}\mathbf{\right(}\mathbf{s}\mathbf{\left)}\mathbf{\right\}}$.

$\mathbf{2}{\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}{\mathbf{ty}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{10}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems1-8use Theorem 7.4.1to evaluate the given Laplace transform.

$\mathbf{}\mathbf{L}\left\{{\mathrm{te}}^{-10t}\right\}$

In Problems 1-8use Theorem 7.4.1to evaluate the given Laplace transform.

$\mathbf{L}\left\{t^3{e}^t\right\}$

Use (8) to evaluate inverse Laplace transform.

${\mathbf{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{s^3\left(s-1\right)}\right\}$

Use (8) to evaluate inverse Laplace transform.

${\mathbf{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{s{\left(s-a\right)}^2}\right\}$

The table in Appendix C does not contain an entry for

${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{8k^{3}s}{{(s^2+k^2)}^3}\right\}$

(a) Use (4) along with the results in (5) to evaluate this inverse transform. Use a CAS as an aid in evaluating the convolution integral.

(b) Re-examine your answer to part (a). Could you have obtained the result in a different manner?

In Problems 1-8use Theorem7.4.1to evaluate the given Laplace transform.

$\mathbf{L}\mathbf{\left\{}\mathbf{tcos}\mathbf{2}\mathbf{t}\mathbf{\right\}}$

Use the Laplace transform and the results of Problem 39 to solve the initial-value problem,

$\begin{array}{l}{\mathbf{y}}^{\mathbf{n}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{sint}\mathbf{+}\mathbf{tsin}\mathbf{;}\\ \mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\end{array}$

Use a graphing utility to graph the solution.