The Unique Solutions to Linear Nonhomo. Sys. of First Order ODEs

The Unique Solutions to Linear Nonhomogeneous Systems of First Order ODEs

Theorem 1: Let $\mathbf{x}' = A(t)\mathbf{x} + g(t)$ with $\mathbf{x}(\tau) = \xi$ be a linear nonhomogeneous system of first order ODEs where $(\tau, \xi) \in D$, and let $\Phi (t, \tau)$ be the state transition matrix of the corresponding linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ with $\mathbf{x}(\tau) = \xi$. Then the unique solution to the linear nonhomogeneous system is given by $\displaystyle{\varphi(t) = \Phi(t, \tau) \xi + \int_{\tau}^{t} \Phi (t, \tau) g (\eta) d \eta}$.

• Proof: Let $\displaystyle{\varphi(t) = \Phi(t, \tau) \xi + \int_{\tau}^{t} \Phi (t, \tau) g (\eta) \: d \eta}$. Then we have that:

(1)

\begin{align} \quad \varphi'(t) &= \left [ \Phi(t, \tau) \xi + \int_{\tau}^{t} \Phi (t, \tau) g (\eta) \: d \eta \right ]' \\ &= \frac{d}{dt} \Phi(t, \tau) \xi + \frac{d}{dt} \int_{\tau}^{t} \Phi(t, \tau) g(\eta) \: d \eta \\ &= \Phi'(t, \tau) \xi + Ig(t) + \int_{\tau}^t \Phi'(t, \eta) g(\eta) \: d \eta \\ &= A(t)\Phi(t, \tau) \xi + g(t) + \int_{\tau}^{t} A(t)\Phi(t, \eta) g(\eta) \: d \eta \\ &= A(t) \left [ \Phi (t, \tau) \xi + \int_{\tau}^{t} \Phi(t, \eta) g(\eta) \: d \eta \right ] + g(t) &= A(t) \varphi(t) + g(t) \end{align}

• So $\varphi$ is a solution to $\mathbf{x}' = A(t)\mathbf{x} + g(t)$. We also have that:

(2)

\begin{align} \quad \varphi (\tau) &= \Phi(\tau, \tau) \xi + \int_{\tau}^{\tau} \Phi (t, \tau) g (\eta) d \eta \\ &= I \xi + 0 \\ &= \xi \end{align}

• So $\varphi$ is a solution to the linear nonhomogeneous system. $\blacksquare$