Solve by using Laplace Transform and explain the steps in brief displaystylefrac{{{d}^{2}{x}}}{{{left.{d}{t}right.}^{2}}}+{2}frac{{{left.{d}{x}right.}}}{{{left.{d}{t}right.}}}+{x}={3}{t}{e}^{{-{t}}} Given x(0)=4, x'(0)=2

Falak Kinney 2020-12-17 Answered
Solve by using Laplace Transform and explain the steps in brief
d2xdt2+2dxdt+x=3tet
Given x(0)=4,x(0)=2
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Expert Answer

Elberte
Answered 2020-12-18 Author has 95 answers
Step 1
The given differential equation is:
d2xdt2+2dxdt+x=3tet(1)
Given: x(0)=4,x(0)=2
Step 2
Taking Laplace transform on both sides of (1). we have,
L{d2xdt2}+2L{dxdt}+L{x}=3L{tet}
x2L(x)xx(0)x(0)+2L(x)2x(0)+L(x)=3(x+1)2
x2L(x)4x2+2[L(x)4]+L(x)=3(x+1)2
L(x){x2+2x+1}=3(x+1)2+4x+10
L(x)=3(x+1)4+4x+10(x+1)2
Step 3
Now by taking inverse Laplace transformation,
L1{L(x)}=3L1{1(x+1)4}+L1{4x+10(x+1)2}
L1{L(x)}=3L1{1(x+1)4}+L1{4x+10(x+1)2}
x=3L1{1(x+1)4}+L1{4x+10(x+1)2}
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