Find the unilateral Laplace transform of the solution to the initial value problem d 202y(t) + 21 y(t) + 4y(t) = 3 cos(2t) for t≥ 0, dt where y(0) = 0 and y'(0) = 1. Find the partial fractions expansion of the unilateral Laplace transform of the solution. Use the table of unilateral Laplace transform pairs to invert the unilateral Laplace transform and determine the solution of the initial value problem. Show that the solution you found satisfies the differential equation and the initial conditions
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
Hello. I am getting lost working through this practice problem in my book. I am very confused and was hoping you could help me out. Thank you for your time.
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