1. Lengkapi hasil transformasi Laplace fungsi pada tabel berikut ini. Transformasi Laplace dari fungsi $f(t)$ didefinisikan sebagai $$F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) dt$$ Beberapa transformasi dasar yang sering digunakan: - $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$ - $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$ - $\mathcal{L}\{\cos(bt)\} = \frac{s}{s^2 + b^2}$ - $\mathcal{L}\{\sin(bt)\} = \frac{b}{s^2 + b^2}$ Mari kita hitung satu per satu: 1.a. $f(t) = 3t$ $$F(s) = 3 \cdot \mathcal{L}\{t\} = 3 \cdot \frac{1}{s^2} = \frac{3}{s^2}$$ 1.b. $f(t) = 2t^3$ $$F(s) = 2 \cdot \mathcal{L}\{t^3\} = 2 \cdot \frac{3!}{s^{4}} = 2 \cdot \frac{6}{s^{4}} = \frac{12}{s^{4}}$$ 1.c. $f(t) = 3e^{2t}$ $$F(s) = 3 \cdot \mathcal{L}\{e^{2t}\} = 3 \cdot \frac{1}{s-2} = \frac{3}{s-2}$$ 1.d. $f(t) = \cos 3t$ $$F(s) = \frac{s}{s^2 + 3^2} = \frac{s}{s^2 + 9}$$ 1.e. $f(t) = \sin 5t$ $$F(s) = \frac{5}{s^2 + 5^2} = \frac{5}{s^2 + 25}$$ 2. Lengkapi hasil invers transformasi Laplace pada tabel berikut ini. Invers transformasi Laplace didefinisikan sebagai $$f(t) = \mathcal{L}^{-1}\{F(s)\}$$ Beberapa invers dasar: - $\mathcal{L}^{-1}\{\frac{n!}{s^{n+1}}\} = t^n$ - $\mathcal{L}^{-1}\{\frac{1}{s-a}\} = e^{at}$ - $\mathcal{L}^{-1}\{\frac{s}{s^2 + b^2}\} = \cos bt$ - $\mathcal{L}^{-1}\{\frac{b}{s^2 + b^2}\} = \sin bt$ Mari kita hitung satu per satu: 2.a. $F(s) = \frac{10}{s^2}$ $$f(t) = 10 \cdot \mathcal{L}^{-1}\{\frac{1}{s^2}\} = 10t$$ 2.b. $F(s) = \frac{12}{s-7}$ $$f(t) = 12 \cdot \mathcal{L}^{-1}\{\frac{1}{s-7}\} = 12 e^{7t}$$ 2.c. $F(s) = \frac{s}{s^2 + 49}$ $$f(t) = \cos 7t$$ 2.d. $F(s) = \frac{7}{s^2 + 49}$ $$f(t) = \frac{7}{7} \sin 7t = \sin 7t$$ 2.e. $F(s) = \frac{6}{s^4}$ Gunakan rumus $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$, maka inversnya: $$\frac{6}{s^4} = \frac{3!}{s^4} \Rightarrow f(t) = t^3$$ Jadi, hasil lengkapnya: | f(t) | F(s) | |------------|---------------| | a. 3t | $\frac{3}{s^2}$ | | b. 2t^3 | $\frac{12}{s^4}$ | | c. 3e^{2t} | $\frac{3}{s-2}$ | | d. cos 3t | $\frac{s}{s^2 + 9}$ | | e. sin 5t | $\frac{5}{s^2 + 25}$ | | F(s) | f(t) | |----------------|-------------| | a. $\frac{10}{s^2}$ | $10t$ | | b. $\frac{12}{s-7}$ | $12 e^{7t}$ | | c. $\frac{s}{s^2 + 49}$ | $\cos 7t$ | | d. $\frac{7}{s^2 + 49}$ | $\sin 7t$ | | e. $\frac{6}{s^4}$ | $t^3$ |