Questão - Integral e Transformada de Laplace - Equações Diferenciais a 2022 1

Use a transformada de Laplace para determinar a solução da equação integral 9t + \frac{8}{3} \int_0^t \cos(\alpha(t - \tau))y(\tau)d\tau = y(t), para \alpha > 10^{50}. Escolha uma opção: a. \quad y(t) = 24 \frac{1}{\alpha^3} + 9t - 24 \frac{1}{\alpha^2} e^{1} \cos(\sqrt{\alpha^2 - \frac{16}{9}} \ t) + 24 \frac{1}{\alpha^3} \frac{1}{\sqrt{\alpha^2 - \frac{16}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t). b. \quad y(t) = 24 \frac{1}{\alpha^2} + 9t - 24 \frac{1}{\alpha^2} e^{1} \cos(\sqrt{\alpha^2 - \frac{16}{9}} \ t) + 32 \frac{1}{\alpha^3} \frac{1}{\sqrt{\alpha^2 - \frac{16}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t). c. \quad y(t) = 24t + 9t - 9 \frac{1}{\alpha} e^{1} \cos(\sqrt{\alpha^2 - \frac{16}{9}} \ t). d. \quad y(t) = 24 \frac{1}{\alpha^3} + 9t \frac{1}{\sqrt{\alpha^2 - \frac{16}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t). e. \quad y(t) = 24 \frac{1}{\alpha^2} + 9t + 32 \frac{1}{\alpha^2} \frac{1}{\sqrt{\alpha^2 - \frac{16}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t). f. \quad y(t) = 24 \frac{1}{\alpha^2} + 9t - 9 \frac{1}{\alpha} e^{1} \cos(\sqrt{\alpha^2 - \frac{16}{9}} \ t) + 32 \frac{1}{\alpha^2} \frac{1}{\alpha^2} \frac{1}{\sqrt{\alpha^2 - \frac{16}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t). g. \quad y(t) = 24 \frac{1}{\alpha^3} + 9t - 24 \frac{1}{\alpha^2} e^{1} \cos(\sqrt{\alpha^2 - \frac{16}{9}} \ t) + 32 \frac{1}{\alpha^3} \frac{1}{\sqrt{\alpha^2 - \frac{16}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t). h. \quad y(t) = 24 \frac{1}{\alpha^3} + 9t + 24 \frac{1}{\alpha^3} \frac{1}{\sqrt{\alpha^2 - 16 \frac{9}{9}}} e^{1} \sin(\sqrt{\alpha^2 - \frac{16}{9}} \ t).