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: \(\Phi(t) = \mathbf{u}(t) + \mathbf{v}(t).\)