Systems of first order ODE's

Definition

Suppose that $x_1\dots x_n$ are functions of $t$. A set of $n$ differential equations of the form

$$x_i^{\prime }=f_i(x_1\dots x_n,t)$$

is called a system of first order ODE’s. In all these equations $x_i^{\prime }=\frac{d{x}_i}{dt}$.

A solution of a system of first order ODE’s is an $n$-tuple functions $(x_1(t)\dots x_n(t))$. Note that one cannot simply evaluate each function separately since each depend on others.

We call an initial condition a specification of each function $x_i(t_0)=x_{i,0}$.

Check Existence-uniqueness for systems of first order ODE’s for the conditions.

Linear systems

Recall that a first order ODE is said to be linear if it can be written as $y^{\prime }+p(t)y=q(t)$. See Linear equations.

Definition

A system of first order ODE’s said to be linear if it can be written in the form of

\begin{align*} x_{1}' &= a_{11}(t)x_{1} + a_{12}(t)x_{2} + \dots + a_{1n}(t)x_{n} + b_{1}(t) \\ x_{i}' &= a_{i1}(t)x_{1} + a_{i2}(t)x_{2} + \dots + a_{in}(t)x_{n} + b_{i}(t) \\ x_{n}' &= a_{n1}(t)x_{1} + a_{n2}(t)x_{2} + \dots + a_{nn}(t)x_{n} + b_{n}(t)

\end{align*}

for some functions $a_{ij}(t)$ and $b_{i}(t)$. Furthermore if $b_{i}(t) = 0$ for $\forall i$ then system is called a **homogeneous system**. Check [[Existence-uniqueness theorem for differential equations#existence-uniqueness-for-systems-of-first-order-odes|Existence-uniqueness for systems of first order ODE's]] for the conditions.