In Problems , solve the given initial value problem using the method of Laplace transforms
The initial value for is
Applying the Laplace transform and using its linearity we get
Solve for the transform as:
Solve for the partial fraction as:
Solve further as:
Using ,respectively, gives
Therefore, the equation is:
Using the inverse Laplace transform we obtain the solution of given differential equation.
Therefore, the solution of the initial value problem is:
Therefore, the initial value for is
The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y(t) to the Laplace transform of the input function g(t), when all initial conditions are zero. If a linear system is governed by the differential equation
use the linearity property of the Laplace transform and Theorem 5 on page363 on the Laplace transform of higher-order derivatives to determine the transfer function of this system.
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