Find Solution (1 point) Consider the initial value problem y? + 19y = cos(71), y

Question

Find Solution

(1 point) Consider the initial value problem y? + 19y = cos(71), y(0) = 5, y?(0) = 3. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). help (formulas) b. Solve your equation for Y(s). Y(s) = {y(t)} = C. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t).

Explanation / Answer

a) We want to apply the Laplace transform to the following differential equation:
y'' + 49 y = cos(7t),
with the initial conditions y(0) = 5 and y'(0) = 3.
We remember that the Laplace transform L() of the functions involved in the equation are:
L(y''(t)) = s^2 F(s) - s y(0) - y'(0),
L(y(t)) = F(s),
L(cos(7t)) = s/(s^2+7^2).
With this, the differential equation transforms to a algebraic equation given by
s^2 F(s) - s y(0) - y'(0) + 49 F(s) = s/(s^2+7^2),
and after replacing the numerical values of the initial conditions, it transforms to
s^2 F(s) - 5 s - 3 + 49 F(s) = s/(s^2+7^2),

b) We solve for F(s), obtaining
(s^2 + 49 ) F(s) - 5 s - 3 = s/(s^2+7^2),
(s^2 + 49 ) F(s) = s/(s^2+7^2)+ 5 s + 3,
F(s) = s/(s^2+7^2)^2 + 5s/(s^2+7^2) + 3/(s^2+7^2).

c) We use the inverse Laplace transform in order to find y(t), denoting it by L^-1, we have that
L^-1 ( s/(s^2+7^2)^2 ) = L^-1 ( 2*7*s/(s^2+7^2)^2 )/14 = t*sin(7t)/14,
L^-1 ( 5s/(s^2+7^2) ) = 5 cos(7t),
L^-1 ( 3/(s^2+7^2) ) = L^-1 ( 7/(s^2+7^2) )*3/7 = 3*sin(7t) /7,
Therefore, y(t) is given by
y(t) = t*sin(7t)/14 + 5 cos(7t) + 3*sin(7t) /7.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.