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To this point, we have considered IVPs with 1st-order

Consider an nth order IVP which consists of an nth order ODE and n initial conditions:

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)}

For example,

y⁽²⁾(t) = 4 − sin(t) + y(t) − 2 y⁽¹⁾(t)
y(0) = 1
y⁽¹⁾(0) = 2

y⁽³⁾(t) = y(t) − t y⁽¹⁾(t) + 4 y⁽²⁾(t)
y(2) = 1
y⁽¹⁾(2) = 2
y⁽²⁾(2) = 3

Suppose we want to find the estimate y₁ of y(t₁). To do this we will convert the nth-order IVP into a system of n 1st-order IVPs by defining:

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

We note that for i = 0, 1, ..., n − 2, the derivative y_i⁽¹⁾(t) = y_i + 1⁽¹⁾(t) and that y_n - 1⁽¹⁾(t) = f( t, y₀(t), y₁(t), y₂(t), ..., y_n − 1(t) ).

The initial conditions may now be converted to

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

If we now define the vectors:

and

and the vector valued function:

then our initial value problem becomes the following vector-valued initial value problem:

y⁽¹⁾(t) = f( t, y(t) )
y(t₀) = y₀

where the derivative of the vector y(t) is the vector of element-wise derivatives.

Any of the techniques we have seen, Euler's method, Heun's method, 4th-order Runge Kutta, or the backward-Euler's method may be applied to approximate y(t₁).

Given our approximation y₁ = (y_{1, i}) of y(t₁), we find that y(t₁) ≈ y_{1, 1}. We also get, for free, approximations of y⁽¹⁾(t₁) ≈ y_{1, 2}, y⁽²⁾(t₁) ≈ y_{1, 3}, ..., and y^{(n − 1)}(t₁) ≈ y_{1, n}.

Additionally, we may apply any of the multiple-step methods, including Runge Kutta Fehlberg.

Example

If we look at the two problems shown above, we could rewrite the first:

y⁽²⁾(t) = 4 − sin(t) + y(t) − 2 y⁽¹⁾(t)
y(0) = 1
y⁽¹⁾(0) = 2

by defining

where the 2nd-order ODE may now be written as the following system of ODEs

with the initial condition

The second example,

y⁽³⁾(t) = y(t) − t y⁽¹⁾(t) + 4 y⁽²⁾(t)
y(2) = 1
y⁽¹⁾(2) = 2
y⁽²⁾(2) = 3

may be rewritten as a system of three functions:

where the 3rd-order ODE may now be written as the following system of ODEs

with the initial condition

We may now solve either of these systems of IVPs using the techniques explained in the previous topic on systems of IVPs.

Topic 14.8: Higher-Order Initial-Value Problems (Theory)

Example