Answer to Question #297044 in Differential Equations for Rio

Question

<e> a) Establish the relation 𝑑

𝑛

𝑑𝑥

𝑛

(𝑥

2 − 1)

𝑛 = 𝑛! 2

𝑛 𝑃𝑛(𝑥)

b) Use Laplace transform to solve

𝑑

2𝑦

𝑑𝑡

2 + 2

𝑑𝑦

𝑑𝑡 + 𝑦 = 𝑡𝑒

−𝑡 𝑤𝑖𝑡ℎ 𝑦(0) = 1, 𝑦

′

(0) = −2