To find: The solution of the given initial value problem.

Emeli Hagan

Emeli Hagan

Answered question

2020-11-23

To find: The solution of the given initial value problem.

Answer & Explanation

Ezra Herbert

Ezra Herbert

Skilled2020-11-24Added 99 answers

Given:
The system of equation is,
(D4)x+6y=9e3tx(D1)y=5e3t
The initial conditions are given as, x(0)=9y(0)=4
Calculation: LetX(s)=L{x}(s)andY(s)=L{y}(s)
The given differential equation is written as,
x4x+6y=9e3t(1)
xy+y=5e3t(2)
Take Laplace transform on both sides of equation (1) and (2) and apply linear property as,
L{x4x+6y}(s)=L{9e3t}(s)
L{x}(s)4L{x}(s)+6L{y}(s)=9L{e3t}(s)(3)
L{xy+y}(s)=L{5e3t}(s)
L{x}(s)L{y}(s)+L{y}(s)=5L{e3t}(s)(4)
Use Laplace transforms formulas in equation (3) and (4) as,
sX(s)x(0)4X(s)+6Y(s)=9s+3
X(s)sY(s)y(0)+Y(s)=5s+3
Substitute the initial conditions as,
sX(s)+94X(s)+6Y(s)=9s+3
X(s)sY(s)4+Y(s)=5s+3(5)
Simplify the system of equations (5) as,
(s4)X(s)+6Y(s)=9+9s+3(6)

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