Question

Let \(\mathbf{x}=\Phi(t)\) be the general solution of \(\mathbf{x}^{\prime}=\mathbf{P}(t) \mathbf{x}+\mathbf{g}(t),\) and let \(\mathbf{x}=\mathbf{v}(t)\) be some particular solution of the same system. By considering the difference \(\boldsymbol{\phi}(t)-\mathbf{v}(t),\) show that \(\Phi(t)=\mathbf{u}(t)+\mathbf{v}(t),\) where \(\mathbf{u}(t)\) is the general solution of the homogeneous system \(\mathbf{x}^{\prime}=\mathbf{P}(t) \mathbf{x} .\)

Short answer

Question: Show that the general solution of a non-homogeneous linear differential equation system can be expressed as the sum of a particular solution and the general solution of the corresponding homogeneous system. Answer: If \(\Phi(t)\) is the general solution of the non-homogeneous system and \(\mathbf{v}(t)\) is a particular solution, then their difference, \(\mathbf{u}(t) = \Phi(t) - \mathbf{v}(t),\) satisfies the homogeneous system \(\mathbf{u}^{\prime}(t) = \mathbf{P}(t) \mathbf{u}(t).\) Therefore, the general solution of the non-homogeneous system can be expressed as the sum of the general solution of the homogeneous system and a particular solution: \(\Phi(t) = \mathbf{u}(t) + \mathbf{v}(t).\)

Step-by-step solution

Define the Difference

First, let's define the difference between the general solution, \(\mathbf{x}=\Phi(t)\), and the particular solution, \(\mathbf{x}=\mathbf{v}(t)\), as \(\mathbf{u}(t) = \Phi(t) - \mathbf{v}(t).\)

Differentiate the Difference with Respect to t

Next, we shall find the derivative of \(\mathbf{u}(t)\) with respect to t. To do this, differentiate both \(\Phi(t)\) and \(\mathbf{v}(t)\) with respect to t and then subtract: \(\mathbf{u}^{\prime}(t) = \Phi^{\prime}(t) - \mathbf{v}^{\prime}(t).\)

Substituting Known Differential Equations for \(\Phi^{\prime}(t)\) and \(\mathbf{v}^{\prime}(t)\)

Now we can substitute the given differential equation system into our expression for \(\mathbf{u}^{\prime}(t)\). We know that \(\Phi^{\prime}(t) = \mathbf{P}(t) \Phi(t) + \mathbf{g}(t),\) and \(\mathbf{v}^{\prime}(t) = \mathbf{P}(t) \mathbf{v}(t) + \mathbf{g}(t).\) Substituting these expressions into the expression for \(\mathbf{u}^{\prime}(t)\), we get: \(\mathbf{u}^{\prime}(t) = (\mathbf{P}(t) \Phi(t) + \mathbf{g}(t)) - (\mathbf{P}(t) \mathbf{v}(t) + \mathbf{g}(t)).\)

Simplifying the Expression for \(\mathbf{u}^{\prime}(t)\)

Simplifying the expression for \(\mathbf{u}^{\prime}(t)\) gives us: \(\mathbf{u}^{\prime}(t) = \mathbf{P}(t) \Phi(t) - \mathbf{P}(t) \mathbf{v}(t).\) Next, factor out \(\mathbf{P}(t)\): \(\mathbf{u}^{\prime}(t) = \mathbf{P}(t) (\Phi(t) - \mathbf{v}(t)).\)

Connecting \(\mathbf{u}^{\prime}(t)\) and \(\mathbf{u}(t)\)

Recall that we previously defined \(\mathbf{u}(t) = \Phi(t) - \mathbf{v}(t)\). So we can rewrite the expression for \(\mathbf{u}^{\prime}(t)\) as: \(\mathbf{u}^{\prime}(t) = \mathbf{P}(t) \mathbf{u}(t).\) Notice that this differential equation represents the general solution to the homogeneous system.

Concluding the Proof

To sum up, we've shown that if \(\Phi(t)\) is the general solution of the non-homogeneous system and \(\mathbf{v}(t)\) is a particular solution, then their difference, \(\mathbf{u}(t) = \Phi(t) - \mathbf{v}(t),\) satisfies the homogeneous system \(\mathbf{u}^{\prime}(t) = \mathbf{P}(t) \mathbf{u}(t).\) Therefore, the general solution of the non-homogeneous system can be expressed as the sum of the general solution of the homogeneous system and a particular solution: \(\Phi(t) = \mathbf{u}(t) + \mathbf{v}(t).\)

Key concepts

Understanding the General Solution

The general solution is an overarching concept when dealing with differential equations, particularly non-homogeneous ones. Imagine non-homogeneous differential equations as a blend of two parts - the characteristics from the system itself and the influences from external factors. The general solution captures both these elements.

• For a given system such as \[ \mathbf{x}' = \mathbf{P}(t) \mathbf{x} + \mathbf{g}(t), \] the general solution, denoted as \( \Phi(t) \), incorporates the full essence of this equation by including all possible particular solutions that satisfy the equation.
• The notion of the general solution is critical because it provides a complete set of possible behaviors of the system.
• What's unique is that \( \Phi(t) \) represents not just one solution but a family of solutions; this means we are not missing any essential characteristic of the system in question.

Ultimately, the general solution of a non-homogeneous equation consists of a particular solution that satisfies the non-homogeneous part of the system and a general solution to the corresponding homogeneous equation. This union ensures that the general solution fully captures all aspects of the system.

Unveiling the Particular Solution

Finding a particular solution for a non-homogeneous equation involves identifying one specific solution that fits the equation \[ \mathbf{x}' = \mathbf{P}(t) \mathbf{x} + \mathbf{g}(t). \]

• Unlike the general solution, the particular solution zeros in on meeting the exact requirements without spanning a family of solutions.
• It is essential to pinpoint because, when combined with the homogeneous solution, it constructs the most comprehensive representation of the system's behavior.
• For example, if \( \mathbf{v}(t) \) is found as a particular solution to the non-homogeneous equation, it corresponds to one very precise trajectory or state that the system can take.

Determining this unique solution often involves various methods of substitution or applying special conditions to solve the equation directly. Once found, it serves as a crucial building block in deriving the overall solution.

Deciphering the Homogeneous System

The homogeneous system relates to a simplified model where the external forces or non-homogeneous elements disappear. It is described by the equation \[ \mathbf{x}' = \mathbf{P}(t) \mathbf{x}. \]

• This system highlights the internal or inherent dynamics of the differential equation without external influences.
• Understanding the homogeneous system is vital because it provides the general solution \( \mathbf{u}(t) \) that describes the behavior of the system under no external conditions.
• The differential equation for the homogeneous part represents natural oscillations, growth, or decay typical to the system's design.

Interestingly, the insight gained from solving the homogeneous system plays a critical role in crafting the general solution \( \Phi(t) = \mathbf{u}(t) + \mathbf{v}(t) \), where \( \mathbf{u}(t) \) forms the core of all behaviors predicted by the system. Solving this part of the equation requires classical techniques such as finding eigenvalues and eigenvectors, depending on the complexity of \( \mathbf{P}(t) \). By combining this with particular solutions, we embrace the complete spectrum of system responses.