Question

Consider the
differential equation
*y*′′(*t*)+4*y*′(*t*)+5*y*(*t*)=74exp(−8t),

with initial conditions *y*(0)=12, and
*y*′(0)=−44.

A)Find the Laplace
transform of the solution *Y*(*s*).Y(s). Write the
solution as a single fraction in *s.*

*Y(s)=
______________*

B) Find the partial
fraction decomposition of Y(s). Enter all factors as first order
terms in *s*, that is, all terms should be of the form
(c/(s-p)), where c is a constant and the root p is a constant. Both
c and p may be complex.

Y(s)= ____ + ______ +______

C) Find the inverse transform of Y(s). The solution must consist of all real terms.

L−1{*Y*(*s*)} = _______________________

Answer #1

Use the Laplace transform to solve the following initial value
problem
y”+4y=cos(8t)
y(0)=0, y’(0)=0
First, use Y for the Laplace transform of y(t) find the
equation you get by taking the Laplace transform of the
differential equation and solving for Y:
Y(s)=?
Find the partial fraction decomposition of Y(t) and its
inverse Laplace transform to find the solution of the IVP:
y(t)=?

Use the Laplace transform to solve the following, given the
initial conditions: y^'' +5y^'+4y = 0 y(0)=1,y^' (0)=0.

Consider the initial value problem
y′′+4y=16t,y(0)=8,y′(0)=6.y″+4y=16t,y(0)=8,y′(0)=6.
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).
Solve your equation for Y(s)
Y(s)=L{y(t)}=__________
Take the inverse Laplace transform of both sides of the
previous equation to solve for y(t)y(t).
y(t)=__________

Solve the initial value problem using the method of
the laplace transform.
y"+4y'+4y=8t,y(0)=-4,y'(0)=4

Consider the following initial value problem.
y′ + 5y =
{
0
t ≤ 2
10
2 ≤ t < 7
0
7 ≤ t < ∞
y(0) = 5
(a)
Find the Laplace transform of the right hand side of the above
differential equation.
(b)
Let y(t) denote the solution to the above
differential equation, and let Y((s) denote the
Laplace transform of y(t). Find
Y(s).
(c)
By taking the inverse Laplace transform of your answer to (b),
the...

Use the Laplace transform to solve the following initial value
problem:
y′′ + 8y ′+ 16y = 0
y(0) = −3 , y′(0) = −3
First, using Y for the Laplace transform of y(t)y, i.e., Y=L{y(t)},
find the equation you get by taking the Laplace transform of the
differential equation
__________________________ = 0
Now solve for Y(s) = ______________________________ and write the
above answer in its partial fraction decomposition, Y(s) = A /
(s+a) + B / ((s+a)^2)
Y(s) =...

find the differential equation using laplace transform.
y'' + 4y = 15et

Use partial fraction decomposition to find the inverse Laplace
transform of the given function.
(a) Y (s) = 2 /(s 2+3s−4)
(b) Y (s) = 1−2s /(s 2+4s+5)
differential eq

Use the Laplace Transform method to solve the following
differential equation problem: y 00(t) − y(t) = t + sin(t), y(0) =
0, y0 (0) = 1
Please show partial fraction steps to calculate
coeffiecients.

Given the differential equation
y''−2y'+y=0, y(0)=1, y'(0)=2
Apply the Laplace Transform and solve for Y(s)=L{y}
Y(s) =
Now solve the IVP by using the inverse Laplace Transform
y(t)=L^−1{Y(s)}
y(t) =

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