Problem

Given the IVP

y⁽ⁿ⁾(t) = f( t, y(t), y⁽¹⁾(t), y⁽²⁾(t), ..., y^{(n − 1)}(t) )
y(t₀) = y₀
y⁽¹⁾(t₀) = y₀⁽¹⁾
y⁽²⁾(t₀) = y₀⁽²⁾
⋅
⋅
⋅
y^{(n − 1)}(t₀) = y₀^{(n − 1)}

approximate y(t₁).

Assumptions

The function f(t, y, y(t), y⁽¹⁾(t), y⁽²⁾(t), ..., y^{(n − 1)}(t)) should be continuous in all variables.

We will convert the nth-order IVP into a system of 1st-order IVPs and then use the techniques we have seen for IVPs.

Define:

y₀(t) = y(t)
y₁(t) = y⁽¹⁾(t)
y₂(t) = y⁽²⁾(t)
⋅
⋅
y_{n − 1}(t) = y^{(n − 1)}(t)

and

y₀(t₀) = y₀
y₁(t₀) = y₀⁽¹⁾
y₂(t₀) = y₀⁽²⁾
⋅
⋅
y_{n − 1}(t₀) = y₀^{(n − 1)}

Next, define the vector-valued function

the vector-valued function

and the initial vector

we may now write the nth-order IVP as the 1st-order IVP

y⁽¹⁾(t) = f( t, y(t) )
y(t₀) = y₀

We may now use all the techniques we have used for 1st-order IVPs, only now, to vectors instead of scalars.