Existence and Uniqueness Theorem

First Order

Theorem

• Given
  • The IVP $\frac{dy}{dx}=f(x,y),y(x_0)=y_0$
• Let
  • $R=\{(x,y)\mid a<x<b,c<y<d\}$ that contains $(x_0,y_0)$
  • $a,b$ can be arbitrarily close to $x_0$ and same for $c,d,y_0$
• If
  • $f$ is continuous in R
  • $\frac{\partial f}{\partial y}$ is continuous in R
• Then
  • The IVP has a unique solution $\varphi (x)$ in some interval $x_0-\delta <x<x_0+\delta$, where $\delta \ge 0,\delta \in [a,b]$

Application

• The IVP $\frac{dy}{dx}=f(x,y),y(x_0)=y_0$ if
  • $f(x,y)$ is continuous near $(x_0,y_0)$
  • $f_y^{\prime }(x,y)$ is continuous near $(x_0,y_0)$

Higher Order

• Given
  • $y^{\prime \prime }(t)+p(t)y^{\prime }(t)+q(t)y(t)=g(t)$
  • $y(t_0)=Y_0,y^{\prime }(t_0)=Y_1$
• If
  • $p(t)$, $q(t)$, $g(t)$ are continuous on an interval $(a,b)$ that contains point $t_0$
• Then
  • For any choice of the initial values $Y_0$ and $Y_1$ there exists a unique solution $y(t)$ on the same interval $(a,b)$ to the initial value problem