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Q7.3 - 17E

Expert-verified
Found in: Page 365

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# 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.]${\mathbf{sin}}{\mathbf{2}}{\mathbf{t}}{\mathbf{}}{\mathbf{sin}}{\mathbf{5}}{\mathbf{t}}$

The Laplace transform for the given equation is $\frac{20s}{\left({s}^{2}+9\right)\left({s}^{2}+49\right)}$.

See the step by step solution

## Definition of Laplace transform

• The integral transform of a given derivative function with real variable t into a complex function with variable s is known as the Laplace transform.
• Let f(t) be supplied for t(0), and assume that the function meets certain constraints that will be presented subsequently.
• The Laplace transform formula defines the Laplace transform of f(t), which is indicated by $\mathcal{L}\left\{f\left(t\right)\right\}$ or F(s).

## Determine the Laplace transform for the given equation

Given that $\mathrm{sin}2t\mathrm{sin}5t$,

Find the Laplace transform of $\mathrm{sin}2t\mathrm{sin}5t$ using ${s}{i}{n}{a}{s}{i}{n}{b}{=}\frac{1}{2}{\left[}{c}{o}{s}{\left(}{a}{-}{b}{\right)}{-}{c}{o}{s}{\left(}{a}{+}{b}{\right)}{\right]}$, ${c}{o}{s}{\left(}{-}{x}{\right)}{=}{c}{o}{s}{x}$, ${\mathcal{L}}{\left\{}{a}{f}{\left(}{x}{\right)}{±}{b}{g}{\left(}{x}{\right)}{\right\}}{=}{a}{\mathcal{L}}{\left\{}{f}{\right\}}{±}{b}{\mathcal{L}}{\left\{}{g}{\left(}{t}{\right)}{\right\}}$, ${\mathcal{L}}{\left\{}{c}{o}{s}{b}{t}{\right\}}{=}\frac{s}{{s}^{2}+{b}^{2}}$ and $\frac{a}{c}{±}\frac{b}{d}{=}\frac{da±cb}{cd}$as:

$\begin{array}{rcl}\mathcal{L}\left\{\mathrm{sin}2t\mathrm{sin}5t\right\}& =& \mathcal{L}\left\{\frac{1}{2}\left[\mathrm{cos}\left(2t-5t\right)-\mathrm{cos}\left(2t+5t\right)\right]\right\}\\ & =& \frac{1}{2}\left[\mathcal{L}\left\{\mathrm{cos}3t\right\}-\mathcal{L}\left\{\mathrm{cos}7t\right\}\right]\\ & =& \frac{1}{2}\left[\frac{s}{{s}^{2}+9}-\frac{s}{{s}^{2}+49}\right]\\ & =& \frac{1}{2}\left[\frac{\left({s}^{2}+49\right)×s-\left({s}^{2}+9\right)×s}{\left({s}^{2}+9\right)\left({s}^{2}+49\right)}\right]\end{array}$

Simplify the equation as:

$\begin{array}{rcl}\mathcal{L}\left\{\mathrm{sin}2t\mathrm{sin}5t\right\}& =& \frac{1}{2}\left[\frac{{s}^{3}+49s-{s}^{3}-9s}{\left({s}^{2}+9\right)\left({s}^{2}+49\right)}\right]\\ & =& \frac{1}{2}\left[\frac{40s}{\left({s}^{2}+9\right)\left({s}^{2}+49\right)}\right]\\ & =& \frac{20s}{\left({s}^{2}+9\right)\left({s}^{2}+49\right)}\end{array}$

Therefore, the Laplace transform for the given equation is $\frac{20s}{\left({s}^{2}+9\right)\left({s}^{2}+49\right)}$.