Please show work so I can learn Solve the initial value problem of the system y1

Question

Please show work so I can learn

Solve the initial value problem of the system
                y1' = 3y1 - 3y2
                y'2 = 3y1 + 3y2
                given y1(0) = 0, y2(0) = 2.Verify the result using MATLAB.

                a. y1 = 2e^(3t)sin(3t),
                    y2 = -2e^(3t)cos(3t),
                
                b. y1 = -2e^(3t)sin(3t),
                    y2 = 2e^(3t)cos(3t)
                
                c. y1 = -2e^(-3t)sin(3t),
                    y2 = 2e^(-3t)cos(3t)
                
                d. y1 = -2e^(3t)cos(3t),
                    y2 =  2e^(3t)sin(3t)

Solve the initial value problem of the system y1' = 3y1 - 3y2 y'2 = 3y1 + 3y2 given y1(0) = 0, y2(0) = 2.Verify the result using MATLAB. y1 = 2e^(3t)sin(3t), y2 = -2e^(3t)cos(3t), y1 = -2e^(3t)sin(3t), y2 = 2e^(3t)cos(3t) y1 = -2e^(-3t)sin(3t), y2 = 2e^(-3t)cos(3t) y1 = -2e^(3t)cos(3t), y2 = 2e^(3t)sin(3t)

Explanation / Answer

L ( y') = SY - y(0)

on applying Laplace's transform,

SY1 - 0 = 3Y1 - 3Y2

SY2 - 2 = 3Y1 + 3Y2

FROM 1ST EQN,

Y1 = -3Y2 / (S-3)

SUBSTITUTE IN EQN2

SY2 - 2 = -9Y2/(S-3) + 3Y2

Y2 = 2 / [ (S-3)^2 + 9 ]/ (S-3)

= 2(S-3) / [ (S-3)^2 + 9 ]

i.e y2 (t) = 2e^(3t)cos(3t)

FROM 2nd EQN,

Y2 = [ 3Y1 + 2 ] / (S-3)

SUBSTITUTE IN EQN1

SY1 - 0 = 3Y1 - { 9Y1 + 6} / (S-3)

Y1 = - 6/ [(S-3)^2 + 9] (-6 = -2 X 3)

i.e y1(t) = -2e^(3t)sin(3t)

i.e option B

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