Question

2) Solve the following initial value problem by using Laplace Transformation. You may use the table given below. y'' - y = t, y(0) = 1, y'(0) = 1 . (f(t)) L(f(t)) 1 1 1/s 2 t 1/s^2 3 t^2 2/s^3 4 t^n (n = 0, 1, ...) n!/s^(n+1) 5 t^a (a positive) ?(a + 1)/s^(a+1) 6 e^(at) 1/(s - a) 7 cos ?t s/(s^2 + ?^2) 8 sin ?t ?/(s^2 + ?^2) 9 cosh at s/(s^2 - a^2) 10 sinh at a/(s^2 - a^2) 11 e^(at) cos ?t (s - a)/((s - a)^2 + ?^2) 12 e^(at) sin ?t ?/((s - a)^2 + ?^2)

          2) Solve the following initial value problem by using Laplace Transformation. You may use the table given below.

y'' - y = t, y(0) = 1, y'(0) = 1 .

(f(t)) L(f(t))
1 1 1/s
2 t 1/s^2
3 t^2 2/s^3
4 t^n (n = 0, 1, ...) n!/s^(n+1)
5 t^a (a positive) ?(a + 1)/s^(a+1)
6 e^(at) 1/(s - a)
7 cos ?t s/(s^2 + ?^2)
8 sin ?t ?/(s^2 + ?^2)
9 cosh at s/(s^2 - a^2)
10 sinh at a/(s^2 - a^2)
11 e^(at) cos ?t (s - a)/((s - a)^2 + ?^2)
12 e^(at) sin ?t ?/((s - a)^2 + ?^2)

2) Solve the following initial value problem by using Laplace Transformation. You may use the table given below.

y” - y = t, y(0) = 1, y'(0) = 1 .

(f(t)) L(f(t))
1 1 1/s
2 t 1/s^2
3 t^2 2/s^3
4 t^n (n = 0, 1, ...) n!/s^(n+1)
5 t^a (a positive) ?(a + 1)/s^(a+1)
6 e^(at) 1/(s - a)
7 cos ?t s/(s^2 + ?^2)
8 sin ?t ?/(s^2 + ?^2)
9 cosh at s/(s^2 - a^2)
10 sinh at a/(s^2 - a^2)
11 e^(at) cos ?t (s - a)/((s - a)^2 + ?^2)
12 e^(at) sin ?t ?/((s - a)^2 + ?^2)

Added by David G.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Zhiqiao Guo

02/01/2025

Step 1

First, we take the Laplace Transform of the given differential equation: $$\mathcal{L}\{y''(t) - y(t) = t\}$$ Using the properties of Laplace Transform, we have: $$\mathcal{L}\{y''(t)\} - \mathcal{L}\{y(t)\} = \mathcal{L}\{t\}$$ Now, we use the initial Show more…

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