Given dydt 4 y 40 sin3t Initial Condition y0 6 Find yt us

Given: dy/dt + 4 y = 40 sin)3t), Initial Condition: y(0) = 6 Find y(t) using The Laplace Transform Method (no other method is allowed)

Solution

y\' + 4y = 40sin(3t)

Taking Laplace Transform on both the sides and let L[y] = Y

=> L[y\'] + 4L[y] = L[40sin(3t)]

=> sY - y(0) + 4Y = 120/(s² + 9)

Plugging the value of y(0) = 6

=> Y(s + 4) - 6 = 120/(s² + 9)

=> Y(s + 4) = (174 + 6s²)/(s² + 9)

=> Y = (174 + 6s²)/( (s² + 9)(s + 4) )

=> Y = -24s/( 5(s² + 9) ) + 96/( 5(s² + 9) ) + 54/( 5(s + 4) )

Taking Inverse Laplace Transform

=> y(t) = (54/5)e^-4t - (24/5)cos(3t) +(32/5)sin(3t)