Question: Find the Laplace transform of the following functions. Show all details.
#5. f(t) = [e^{2t}] \sinh(t)
#7. f(t) = \sin(\omega t + \theta)
1. **Problem 5:** Find the Laplace transform of $$f(t) = e^{2t} \sinh(t)$$
2. **Recall the Laplace transform definition:**
$$\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) dt$$
3. **Recall the formula for the Laplace transform of $$\sinh(at)$$:**
$$\mathcal{L}\{\sinh(at)\} = \frac{a}{s^2 - a^2}, \quad s > |a|$$
4. **Use the property of the Laplace transform for $$e^{ct} f(t)$$:**
$$\mathcal{L}\{e^{ct} f(t)\} = F(s - c)$$
where $$F(s) = \mathcal{L}\{f(t)\}$$.
5. **Apply this to $$f(t) = e^{2t} \sinh(t)$$:**
Let $$g(t) = \sinh(t)$$, then
$$\mathcal{L}\{g(t)\} = \frac{1}{s^2 - 1}, \quad s > 1$$
Therefore,
$$\mathcal{L}\{e^{2t} \sinh(t)\} = \mathcal{L}\{g(t)\}(s - 2) = \frac{1}{(s - 2)^2 - 1}, \quad s - 2 > 1 \Rightarrow s > 3$$
6. **Simplify the denominator:**
$$ (s - 2)^2 - 1 = s^2 - 4s + 4 - 1 = s^2 - 4s + 3$$
7. **Final answer for problem 5:**
$$\boxed{\mathcal{L}\{e^{2t} \sinh(t)\} = \frac{1}{s^2 - 4s + 3}, \quad s > 3}$$
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8. **Problem 7:** Find the Laplace transform of $$f(t) = \sin(\omega t + \theta)$$
9. **Use the angle addition formula for sine:**
$$\sin(\omega t + \theta) = \sin(\omega t) \cos(\theta) + \cos(\omega t) \sin(\theta)$$
10. **Recall Laplace transforms:**
$$\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$$
$$\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$$
11. **Apply linearity:**
$$\mathcal{L}\{\sin(\omega t + \theta)\} = \cos(\theta) \frac{\omega}{s^2 + \omega^2} + \sin(\theta) \frac{s}{s^2 + \omega^2}$$
12. **Final answer for problem 7:**
$$\boxed{\mathcal{L}\{\sin(\omega t + \theta)\} = \frac{\omega \cos(\theta) + s \sin(\theta)}{s^2 + \omega^2}}$$