Answer
The Laplace transform can be used to solve the given differential equation, 5*(dy/dt) + 4y = 2 and y(0) = 1. This can be done by applying the Laplace transform to both sides of the equation, which will result in a second order linear differential equation with constant coefficients. This can be solved by finding the complementary function and the particular solution.
Laplace Transforming the Equation
The first step in solving this equation is to take the Laplace transform of both sides. For the left side of the equation, the Laplace transform yields 5*s*Y(s)-5*y(0) + 4Y(s) = 5*sY(s) + 4Y(s). For the right side, the Laplace transform yields 2. Therefore, the equation becomes 5*sY(s) + 4Y(s) = 2.
Finding the Complementary Function
The next step is to solve for the complementary function. To do this, we need to rearrange the equation to solve for Y(s). This yields Y(s) = (2/9)*(1/s + 1/3). The complementary function of the equation is then y(t) = (2/9)*e-t/3 + c*e-t/5.
Finding the Particular Solution
The last step is to solve for the particular solution. To do this, we need to use the initial condition, y(0) = 1. This yields 1 = (2/9) + c. Therefore, the particular solution of the equation is y(t) = (2/9)*e-t/3 + (7/9)*e-t/5.
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