Solve the following sets of equations by the Laplace transform method. $$\begin{array}{ll} y^{\prime}-z^{\prime}-y=\cos t & y_{0}=-1 \\ y^{\prime}+y-2 z=0 & z_{0}=0 \end{array}$$

Differential equations are equations that involve an unknown function and its derivatives. They play a crucial role in modeling various physical and engineering systems.

For example, they can describe how heat spreads over time, how populations change, or even how electrical circuits behave.

• Ordinary Differential Equations (ODEs): Involve functions of a single variable and their derivatives.


• Partial Differential Equations (PDEs): Involve functions of multiple variables and their partial derivatives.

Solving differential equations typically involves finding a function that satisfies the equation. This can be done using various methods, such as separation of variables, integrating factors, or transformation methods like the Laplace transform.

In our exercise, we have a system of first-order ODEs with initial conditions.

An initial value problem (IVP) is a type of differential equation along with specified values at the start, known as initial conditions. These conditions are essential because they allow us to find a unique solution out of many possible ones.

For instance, consider the equation: \[ y' - z' - y = \cos t \, \text{with} \, y(0) = -1 \]
Here, we know the value of the function at the start (t=0).

Initial conditions are crucial because they:

• Allow the determination of a specific solution that fits the given conditions.


• Reflect real-world situations where we often know the state of a system at a particular time.

Without initial conditions, we'd only be able to find a general solution with arbitrary constants. The specified initial values allow us to solve for these constants uniquely.

A system of differential equations involves more than one equation and more than one unknown function. They are often used to model complex systems with multiple interdependent components.

For example, the set of equations in our problem involves two functions, y(t) and z(t):

• \(y' - z' - y = \cos t, \, y(0) = -1\)


• \(y' + y - 2z = 0, \, z(0) = 0\)

To solve a system of equations, one common approach is to use methods like substitution, elimination, or transformation (including the Laplace transform).

The Laplace transform method is particularly powerful as it transforms differential equations into algebraic ones, which are often easier to solve. The key steps in this method involve:

• Transforming the differential equations using the Laplace transform.


• Solving the resulting algebraic equations.


• Using the inverse Laplace transform to return to the time domain.

The cosine function, \(\cos t\), appears in the first differential equation of our system: \(y' - z' - y = \cos t\). It is a common trigonometric function that describes oscillatory behavior.

In the context of differential equations, such terms often indicate systems typified by oscillations, such as mechanical vibrations or electrical circuits involving AC currents.

The Laplace transform of the cosine function is particularly useful because it simplifies the equation. The Laplace transform of \(\cos t\) is given by: \[ \mathcal{L}\{ \cos t \} = \frac{s}{s^2 + 1} \]
By using the transform, we convert the trigonometric function into a rational algebraic form that is easier to work with.

The cosine function can greatly impact the nature of the solution. Since it introduces oscillations, the resulting solution will typically involve sinusoidal functions, reflecting the cyclic behavior imposed by the cosine term.

Solving such equations helps us better understand and predict the dynamic behavior of systems influenced by periodic forces.