Use a Laplace transform to determine the solution of the following systems with differential equations a) x' +4x+3y=0 text{ with } x(0)=0 y'+3x+4y=2e^t , y(0)=0

geduiwelh 2021-02-13 Answered
Use a Laplace transform to determine the solution of the following systems with differential equations
a) x+4x+3y=0 with x(0)=0
y+3x+4y=2et,y(0)=0
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Expert Answer

Brighton
Answered 2021-02-14 Author has 103 answers
Solution:
The system of differential equations is
x+4x+3y=0
y+3x+4y=2et
Apply Laplace transform:
sX(s)x(0)+4X(s)+3Y(s)=0(s+4)X(s)+3Y(s)=0
sY(s)y(0)+3X(s)+4Y(s)=2(1(s1))3X(s)+(s+4)Y(s)=2(s1)
Perform substitution:
Y(s)=(s+4)3X(s), so 3X(s)(s+4)23X(s)=2(s1)
X(s)=6(9(s+4)2)(s1), so Y(s)=2(s+4)(9(s+4)2)(s1)
X(s)=6(s+7)(s+1)(s1), so Y(s)=2(s+4)(s+7)(s+1)(s1)
Apply inverse Laplace transform:
Perform partial fractional decomposition:
X(s)=6(s+7)(s+1)(s1)=18(s+7)+12(s+1)38(s1)
Apply inverse Laplace transform:
x(t)=18e7t+12et38et
Perform partial fractional decomposition:
Y(s)=2(s+4)(s+7)(s+1)(s1)=18(s+7)12(s+1)+58(s1)
Apply inverse Laplace transform:
y(t)=18e7t12et+58et
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