Consider the initial value problem for function y
 given by,
y′′+y′−6y=−5e^t,y(0)=0,y′(0)=0.
Find the Laplace Transform of the solution, Y(s)=L[y(t)]
.
Y(s)=(-5)/((s-1)(s^2+s-6))
The function Y(s)
 found in part (a) has a partial fraction decomposition
Y(s)=A/((s−a))+B/((s−b))+C/((s−c)), 
where we assume that a>b>c
. Find the constants A
, B
, C
, a
, b
, and c
.
A=

We are given the Laplace transform \( Y(s) \) of the solution to the differential equation, and our goal is to find the constants \( A \), \( B \), \( C \), \( a \), \( b \), and \( c \) for the partial fraction decomposition of \( Y(s) \).

The form of \( Y(s) \) is given as:
\[
Y(s) = \frac{-5}{(s-1)(s^2 + s - 6)}.
\]
We need to express \( Y(s) \) in terms of partial fractions:
\[
Y(s) = \frac{A}{s - a} + \frac{B}{s - b} + \frac{C}{s - c}.
\]
To do

Consider the initial value problem for function y given by, y′′+y′−6y=−5e^t, y(0)=0, y′(0)=0. Find the Laplace Transform of the solution, Y(s)=L[y(t)]. Y(s)=(-5)/((s-1)(s^2+s-6)). The function Y(s) found in part (a) has a partial fraction decomposition Y(s)=A/((s−a))+B/((s−b))+C/((s−c)), where we assume that a>b>c. Find the constants A, B, C, a, b, and c.

Learn how to find the Laplace transform of a solution to a second-order differential equation and solve for the constants in a partial fraction decomposition. This step-by-step guide covers concepts such as Laplace transforms, partial fractions, and initial value problems. Suitable for college or undergraduate students.

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