Problems using Laplace expansion property to find Laplace Transform.

My problem deals with the Laplace expansion property applied to a exponential product function. I am unsure as to how the problem should be solved based on the given property formula and the Differential Equation books that I have used have been of no help. If familiar, please try to use the DiPrima notation.

Inverse Laplace Transforms and partial fractions

I have a transform F(s) of which I need the inverse transform for. The form of the transform is not of a common form and I am having trouble reducing it to a workable form.

Laplace transform : Solve an Initial Value Problem

Please see the attached file for the fully formatted problem. Suppose there was an IVP such as the following: where Where and how do you begin to set this problem up to be solved using the Laplace transform? The value y(4)(t) is the fourth derivative of function y(t).

General Real-Valued Solutions

Find the general real-valued solution of each system. Classify the origin as a saddle, center, spiral, or one of the normal types (identify the type). Identify as neutrally stable, unstable, or asymptotically stable. *See attachment for systems

Convolution Applied to Inverse Transform Problem

Please see the attached file for the fully formatted problem. My problem lies in manipulating the equation to one which the inverse transform can be taken, but would appreciate this example of convolution worked with some more of the blanks filled. I cannot figure out how the equation gets manipulated into what appears ...continues

Laplace Transform to solve IVP

Use Laplace transform to solve an IVP style problem. Where y^n(t)+a^2y(t)=0 and a is not equal to zero.

Inverse Laplace Transforms f(t) from F(s)

Please see the attached file for the fully formatted problems.

Using definition of Laplace transform, find transform of function f(t)

Simple Laplace transform of a product of two trig identities of which I cannot remember how to integrate.

What is the largest interval on which the solution can be defined?

Please see the attached file for the fully formatted problems. Find the solution y(t) to the initial value problem attached. What is the largest interval on which the solution can be defined?

Find the general solution to the driven differential equation.

Find the general solution to the driven differential equation attached. The solution is detailed and well presented. The solution received a rating of "5" from the student who posted the question.