Solve the following initial value problems by the Laplace...

Solve the following initial value problems by the Laplace transform. (If necessary, use partial fraction expansion as in Example $4 .$ Show all details.)
$$\begin{aligned}
&y^{\prime \prime}+2 y^{\prime}+2 y=0, \quad y(0)=1\\
&y^{\prime}(0)=-3
\end{aligned}$$

Key Concepts

Inverse Laplace Transform

The inverse Laplace transform is the process by which the solution expressed in the Laplace (or s) domain is converted back into the original time domain. This step is necessary after solving the algebraic equation corresponding to the differential equation, and typically involves using established transform tables or additional algebraic techniques such as partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition is an algebraic technique used to break down complex rational functions into simpler fractions. This process is essential in the context of the Laplace transform because it helps to simplify the transformed function into a form where known inverse transforms can be directly applied, facilitating the conversion of the solution back to the time domain.

Laplace Transform

The Laplace transform is an integral transform that converts a function of time into a function of a complex variable, usually denoted s. It is especially useful for solving linear differential equations as it changes differentiation operations into algebraic ones, making it easier to handle initial conditions. Key properties include the transforms of derivatives, which allow the initial value problem to be cast into an algebraic equation.

Initial Value Problem (IVP) for Differential Equations

An initial value problem specifies the value of the unknown function and its derivatives at a certain point, usually at time zero. This additional information is crucial when solving differential equations to determine a unique solution. In the context of Laplace transforms, the initial conditions are embedded into the transformed equation through properties related to the transforms of derivatives.

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