Expert answers to "Find x(t) and Y(t) using Laplace transform. {dxdt=2x−3ydydt=y−2x..."

Mylo O'Moore

Answered question

2020-11-09

Find x(t) and Y(t) using Laplace transform. ${\begin{cases} \frac{d x}{d t} = 2 x - 3 y \frac{d y}{d t} = y - 2 x \end{cases}$
$x (0) = 8, y (0) = 3$

liannemdh

Skilled2020-11-10Added 106 answers

Step 1 The given pair of equations

{\begin{cases} \frac{d x}{d t} = 2 x - 3 y \frac{d y}{d t} = y - 2 x \end{cases}

x (0) = 8, y (0) = 3

Step 2 Take Laplace transform

s x - x (0) = 2 x - 3 y

(s - 2) x = 8 - 3 y

y = \frac{8 - (s - 2) x}{3} . . . . (1)

s y - y (0) = y - 2 x

(s - 1) y = 3 - 2 x . . . . (2)

Put equation (1) in (2) Step 3

(s - 1) \times (\frac{8 (s - 2) x}{3}) = 3 - 2 x

8 (s - 1) - (s - 1) (s - 2) x = 9 - 6 x

((s - 1) (s - 2) - 6) x = 8 (s - 1) - 9

x = \frac{8 s - 17}{s^{2} - 3 s - 4}

x = \frac{8 s - 17}{(s - 4) (s + 1)}

Take partial fractions

x = \frac{3}{s - 4} + \frac{5}{s + 1}

Take inverse Laplace

x (t) = 5 e^{- t} + 3 e^{4 t}

Step 4

\frac{d x}{d t} = 2 x - 3 y

y = \frac{10 e^{- t} + 6 e^{4 t} - (- 5 e^{- t} + 12 e^{4 t})}{3} = \frac{15 e^{- t} - 6 e^{4 t}}{3}

y (t) = 5 e^{- t} - 2 e^{4 t}

Therefore,

x (t) = 5 e^{- t} + 3 e^{4 t}

y (t) = 5 e^{- t} - 2 e^{4 t}

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