Question

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Find the Laplace Transform of the functions x(t) and y(t) , which satisfy the coupled first order differential equations
dx/dt +5x -3y = 0
dx/dt +2 dy/dt = -4t,
subject to the boundary conditions
x(0) = y(0) = 0

Answer 1

To find the Laplace Transform of the functions x(t) and y(t) which satisfy the coupled first order differential equations, we can use the Laplace Transform properties and the initial value theorem. Let's start by taking the Laplace Transform of the given equations and substitute the initial conditions.

Step-by-step explanation:

To find the Laplace Transform of x(t) and y(t) which satisfy the coupled first order differential equations, we can use the Laplace Transform properties and the initial value theorem. Let's start by taking the Laplace Transform of the given equations.

1. Applying the Laplace Transform to the first equation:

   sX(s) - x(0) + 5X(s) - 3Y(s) = 0

   Combining like terms, we get:

   X(s) = (x(0) + 3Y(s))/(s+5)

2. Applying the Laplace Transform to the second equation:

   sX(s) + 2sY(s) - 2y(0) = -4/s^2

   Combining like terms, we get:

   Y(s) = (4/s2 - sX(s))/(2s)

Now we can substitute the initial conditions to find X(s) and Y(s).