Use the Laplace Transform solve the IVP begin{cases}x'+y=t 4x+y'=0end{cases} x(0)=1 y(0)=2

Harlen Pritchard 2021-02-05 Answered
Use the Laplace Transform solve the IVP
\(\begin{cases}x+y=t\
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Expert Answer

d2saint0
Answered 2021-02-06 Author has 89 answers
Step 1
Theory :
The Laplace transform of a second order linear differential equation : y"+αy+βy=f(t) subject to initial conditions y(0)=A and y'(0)=B is given by,
Y(s)=(s+α)A+B(s2+αs+β)+(F(s))(s2+αs+β)
Taking inverse laplace transform on both sides we get, y(t)=L1[(s+α)A+B(s2+αs+β),t]+L1[F(s)(s2+αs+β),t]
Step 2
Given two simultaneous differential equations with ICs,
x+y=t and 4x+y=0 with x(0)=1 and y(0)=2
Taking Laplace transforms on both sides of each equation and using notation Y=L[y,s] we obtain,
sXx(0)+Y=L[t,s]=1s2
sYy(0)+X=L[0,s]=0
Using initial conditions, sX1+Y=1s2
sY2+X=0
sX+Y=1s2(1)
X+sY=2(2)
Solving equations (1) and (2) for X and Y we obtain,
X=12ss(s21)(3)
Y=2ss211s2(s21)(4)
Using partial fractions method in (3) and (4) we further get the values of X and Y as,
X=1s+s(s21)2(s21)
Y=541s+s(s21)
Using inverse laplace transform we get,
x=1+coshtsinht
y=54+cosht
The answer is x=1+coshtsinht
y=54+cosht
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