In Problems , solve the given initial value problem using the method of Laplace transforms.
The Initial value for is
Shift the initial conditions to by defining a new function:
Replace by in the condition
Solve for initial condition.
Solve for differentiation of initial condition.
Simplify the equation as:
sing the properties listed below, take the Laplace transform of the equation
Substitute the properties into the equation.
Substitute the initial conditions:
Distribute and simplify:
Isolate the Y variable.
Find the partial fraction expansion
Because is a repeated factor of ,we include and
Combine the fractions to equate the numerators.
Solve for variables by setting values of S
Substitute the values A,B,C of into partial fraction expansion
Using the properties listed below take the inverse Laplace transform to obtain the solution
Solve for the solution of differential equation as:
Since , , , replace by
Therefore, the Initial value for is
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