MATH528

MATH528 Lesson08: Higher-order linear ODEs

Definition. The general solution of the homogeneous ODE

y^{(n)} + p_{n - 1} (x) y^{(n - 1)} + \dots + p_{1} (x) y^{'} + p_{0} (x) y = 0,

(1)

is $y (x) = c_{1} y_{1} (x) + \dots + c_{n} y_{n} (x)$ with ${y_{1} (x), \dots, y_{n} (x)}$ linearly independent.

Definition. An initial value problem is defined by (1) and $n$ initial conditions

y (x_{0}) = K_{0}, y^{'} (x_{0}) = K_{1}, \dots, y^{(n - 1)} (x_{0}) = K_{n - 1} .

(2)

Theorem. If $p_{i} (x)$ are continuous on $I = (a, b)$ , and $x_{0} \in I$ , then the IVP (1-2) has an unique solution on I.

Theorem. ${y_{1} (x), \dots, y_{n} (x)}$ are linearly dependent iff $\exists x_{0}$ such that

$W (y_{1}, \dots, y_{n}) = | \begin{array}{cccc} y_{1} & y_{2} & \dots & y_{n} \\ y_{1}^{'} & y_{2}^{'} & \dots & y_{n}^{'} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{1}^{(n - 1)} & y_{2}^{(n - 1)} & \dots & y_{n}^{(n - 1)} \end{array} | = 0$

In[1]:= Wronskian[{1,x,x^2,x^3},x]

$12$

In[2]:= Wronskian[{Cos[x], Sin[x], x Cos[x], x Sin[x]},x]

$4$

In[3]:=