We'll start this section off with the material which most text books that will cover in this section. We will take the matter from the Second Order chapter and expand this out to n^th order linear differential equations. As we will see almost all of the 2^nd order material will very naturally extend out to n^th order along with only some bit of new material.

Therefore, let's start things off this time with several fundamental concepts for n^th order linear differential equations. The most general n^th order linear differential equation is as,

P_n(t) y⁽ⁿ⁾ + P_n-1(t) y^(n-1) + ......+ P₁(t) y' + P₀(t) y = G(t)                            (4)

So we hopefully recall this,

y^(m) = d^my/dx^m

Several of the theorems and concepts for this material need that y(n)has a coefficient of 1 and therefore if we divide out by P_n (t) we find,

y⁽ⁿ⁾ + p_n-1(t) y^(n-1) + ......+ p₁(t) y' + p₀(t) y = g(t)                                    (5)

As we may suspect an initial value problem for an n^th order differential equation will need the subsequent n initial conditions.

y(t₀) = y?₀,       y'(t₀) = y?₁,      ........                 y^(n-1) (t₀) = y? _{(n-1) =}  y?_(n-1),        (6)