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Existence and Uniqueness


{\bf 1st order LINEAR differential equation:}


Thm 2.4.1:  If $p:(a, b) \rightarrow R$ and $g:(a, b) \rightarrow R$ are continuous and $a < 
t_0 < b$, then 
there exists a unique function $y = \phi(t)$, $\phi:(a, b) \rightarrow R$ that satisfies the 
initial value problem

\centerline{$y' + p(t) y = g(t)$,} 

\centerline{$y(t_0) = y_0$}



{\bf 2nd order LINEAR differential equation:}


Thm 3.2.1:  If $p:(a, b) \rightarrow R$, $q:(a, b) \rightarrow R$, and $g:(a, b) \rightarrow R$ 
are 
continuous and $a < t_0 < b$, then
there exists a unique function $y = \phi(t)$, $\phi:(a, b) \rightarrow R$ that satisfies the
initial value problem

\centerline{$y'' + p(t) y' + q(t)y = g(t)$,}

\centerline{$y(t_0) = y_0$,}

\centerline{$y'(t_0) = y_0'$}


Thm 3.2.2:  If $\phi_1$ and $\phi_2$ are two solutions to a homogeneous linear differential 
equation, the 
$c_1\phi_1 + c_2\phi_2$ is also a solution to this linear differential equation.


\eject

Definition:  The Wronskian of two differential functions, $f$ and $g$ is 

\centerline{$W(f, g) = fg' - f'g = \big|\matrix{f & g \cr f' & g'}\big|$}

Thm 3.2.3:  Suppose that $\phi_1$ and $\phi_2$ are two solutions to $y'' + p(t) y' + q(t)y = 
0$.  If $W(\phi_1, \phi_2)(t_0) = \phi_1(t_0)\phi_2'(t_0) - \phi_1'(t_0)\phi_2(t_0) \not= 0$, 
then there is a unique choice of constants $c_1$ and $c_2$ such that $c_1\phi_1 + c_2\phi_2$ 
satisfies this 
homogeneous linear differential 
equation and initial conditions, $y(t_0) = y_0$, $y'(t_0) = y_0'$.

Thm 3.2.4:  Given the hypothesis of thm 3.2.1
Suppose that $\phi_1$ and $\phi_2$ are two solutions to $y'' + p(t) y' + q(t)y =
0$.  If $W(\phi_1, \phi_2)(t_0) \not= 0$, 
for some $t_0 \in (a, b)$, then any solution to this homogeneous linear differential
equation can be written as $y = 
c_1\phi_1 + c_2\phi_2$ for some $c_1$ and $c_2$.

Defn  If $\phi_1$ and $\phi_2$ satisfy the conditions in thm 3.2.4, then $\phi_1$ and $\phi_2$ 
form a fundamental set of solutions to $y'' + p(t) y' + q(t)y =
0$. 

Thm 3.2.5:  Given any second order homogeneous linear differential equation, there exist a pair 
of functions which form a fundamental set of solutions.


3.3:  Linear Independence and the Wronskian

Defn: $f$ and $g$ are linearly dependent if there exists constants $c_1, c_2$ 
such that $c_1 \not= 0$ or $c_2 \not= 0$ and $c_1f(t) + c_2g(t) = 0$ for all $t \in (a, 
b)$

Thm 3.3.1:  If $f: (a, b) \rightarrow R$ and $g(a, b) \rightarrow R$ are differentiable 
functions on (a, b) and if $W(f, g)(t_0) \not= 0$ for some $t_0 \in (a, b)$, then $f$ and $g$ 
are linearly independent on $(a, b)$.  Moreover, if $f$ and $g$ are linearly dependent on $(a, 
b)$, then $W(f, g)(t) = 0$  for all $t \in (a, b)$


If $c_1f(t) + c_2g(t) = 0$ for all $t$,
then
$c_1f'(t) + c_2g'(t) = 0$

\vskip 10pt
Solve the following linear system of equations for $c_1, c_2$

$\matrix{c_1 f(t_0) + c_2g(t_0) = 0
\cr
c_1 f'(t_0) + c_2g'(t_0) = 0}$


$\left[\matrix{f(t_0) & g(t_0) \cr
f'(t_0) & g'(t_0)}\right]
\left[\matrix{
c_1 \cr c_2
}\right]
=
\left[\matrix{
0 \cr 0
}\right]$


\end