Existence-uniqueness theorem for differential equations

Existence-uniqueness for systems of first order ODE’s

Theorem

Consider a system of first order ODE’s. $\frac{d{x}_i}{dt}=f_i(x_1\dots x_{n,t})$ together with the initial condition vector $X(t_0)=X_0$. Now assume that

• Each $f_i(x_1\dots x_{n,t})$ and
• Each $\frac{\partial f_i}{\partial x_j}(x_1\dots x_n,t)$ is continuous on a open rectangular box $\in R^{n+1}$ containing $X_0$. Then there is a unique solution of this initial value problem and the solution is valid in some open interval containing $t_0$.

Theorem for linear systems

Consider a system of first order ODE’s. $\frac{d{x}_i}{dt}=f_i(x_1\dots x_{n,t})$ together with the initial condition vector $X(t_0)=X_0$.

\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*}

Now assume that - Each $a_{ij}$ and $b_{i}$ are continuous for $t$ in an open interval $(\alpha,\beta)$. Then there is a unique solution of this initial value problem and the solution is valid over whole interval $(\alpha,\beta)$.