Usar a transformada de Laplace para resolver os problemas de valor inicial abaixo:
Determinar a solução do problema de valor inicial
4 y^{\prime \prime}+4 y^{\prime}+17 y=g(t), \quad y(0)=0,
y^{\prime}(0)=0 em função de uma integral de convolução.
TABLE 6.2.1 Elementary Laplace Transforms
f(t)=\mathcal{L}^{-1}{F(s)} |
F(s)=\mathcal{L}{f(t)} |
Notes |
1.1 |
\frac{1}{s}, \quad s>0 |
Sec. 6.1; Ex. 4 |
2.
e^{a t} |
\frac{1}{s-a}, \quad s>a |
Sec. 6.1; Ex. 5 |
3.
t^{n}; n= positive integer |
\frac{n!}{s^{n+1}}, \quad s>0 |
Sec. 6.1; Prob. 27 |
4.
t^{p}, p>-1 |
\frac{\Gamma(p+1)}{s^{p+1}}, \quad s>0 |
Sec. 6.1; Prob. 27 |
5. sin at |
\frac{a}{s^{2}+a^{2}}, \quad s>0 |
Sec. 6.1; Ex. 6 |
6.
\cos a t |
\frac{s}{s^{2}+a^{2}}, \quad s>0 |
Sec. 6.1; Prob. 6 |
7.
\operatorname{sinh} a t |
(\frac{a}{s{2}-a{2}}, \quad s> |
a |
) |
Sec. 6.1; Prob. 8 |
8. cosh
a t |
(\frac{s}{s{2}-a{2}}, \quad s> |
a |
) |
Sec. 6.1; Prob. 7 |
9.
e^{a t} \sin b t |
\frac{b}{(s-a)^{2}+b^{2}}, \quad s>a |
Sec. 6.1; Prob. 13 |
10.
e^{a t} \cos b t |
\frac{s-a}{(s-a)^{2}+b^{2}}, \quad s>a |
Sec. 6.1; Prob. 14 |
11.
t^{n} e^{a t}, n= positive integer |
\frac{n!}{(s-a)^{n+1}}, \quad s>a |
Sec. 6.1; Prob. 18 |
12.
u_{c}(t) |
\frac{e^{-c s}}{s}, \quad s>0 |
Sec. 6.3 |
13.
u_{c}(t) f(t-c) |
e^{-c s} F(s) |
Sec. 6.3 |
14.
e^{c t} f(t) |
F(s-c) |
Sec. 6.3 |
15.
f(c t) |
\frac{1}{C} F\left(\frac{S}{C}\right), \quad C>0 |
Sec. 6.3; Prob. 19 |
16.
\int_{0}^{t} f(t-\tau) g(\tau) d \tau |
F(s) G(s) |
Sec. 6.6 |
17.
\delta(t-c) |
e^{-c s} |
Sec. 6.5 |
18.
f^{(n)}(t) |
s^{n} F(s)-s^{n-1} f(0)-\cdots-f^{(n-1)}(0) |
Sec. 6.2 |
19.
(-t)^{n} f(t) |
F^{(n)}(s) |
Sec. 6.2; Prob. 28 |