Lecture Notes: Solving Higher Order Equations for Ordinary Differential Equations

Solving Higher Order Equations for Ordinary Differential Equations

Overview

After reading this section, you should be able to:

  1. Solve higher order and coupled differential equations.

We have learned about Euler’s and Runge-Kutta methods to solve first order ordinary differential equations of the form:
rac{dy}{dx} = f(x, y)

Transition to Higher Order and Coupled Differential Equations

The lecture introduces how to approach simultaneous (coupled) differential equations or equations of order higher than first. An example given is an n-th order differential equation of the form:
y^{(n)} = f(x, y, y', y'', …, y^{(n-1)} )
with initial conditions.

System of First Order Differential Equations

To solve this, we can assume a framework that allows us to express the n-th order differential equation in the form of simultaneous first-order differential equations. This can be represented as:

Assumptions:

  • Assume:
    (3.1) \ y1 = y (3.2) \ y2 = y'
    (3.3) \ y_3 = y''

  • Continuing this pattern leads to variables up to:
    (3.n) \ y_n = I(y)

  • Thus, we can set up the equations:
    (3.n+1) \ y_{n+1} = f

These equations from (3.1) to (3.n+1) collectively represent an ensemble of first order differential equations:

Representation as First Order Equations

Using the linear nature of the new variables:
egin{align}
rac{dy1}{dx} &= y2 \
rac{dy2}{dx} &= y3 \
& \vdots \
rac{dyn}{dx} &= f(x, y1, y2, y3, …, y_n) \\
\end{align}
These will each be accompanied by initial conditions.

Example 1: Rewriting Differential Equations

Problem Statement

Rewrite the following differential equation as a set of first order differential equations.

Solution Approach

To demonstrate, we take an ordinary differential equation (ODE) and convert it:

  1. Assume:

  2. Substitute into the original second-order ODE leading to a set of simultaneous first-order differential equations.

  3. Complete with initial conditions.

  4. Now, any numerical methods learned for solving first order ODEs can be applied.

Example 2: Application of Euler’s Method

  • Given an ODE, find:
    a) Solutions using Euler’s method
    b) The absolute relative true error.

Important Values

  • Initial conditions and variables specified.

Computation of Euler's Method:

Using a step size of $h = 0.25$ and computing iteratively:

  • The updates will be shown in sequence from $t=0$ to $t=0.75$ with adjustments made according to the previous states.

  • More specifically:

    • E.g., y{i+1} = yi + f(ti, yi, z_i) h

    • Each state is computed leading to values such as $y(0.75) \approx 1.8299$

Example 3: Utilizing Heun’s Method

  • Following a similar rewrite strategy to Example 2, but changing the method of approach.

Key Equations

  1. Set up the first order equations:
    (E3.1) \ (E3.2)

  2. Use Heun’s method to find predictions of values iteratively:
    egin{align}
    y^{(i+1)} &= yi + rac{1}{2}(k1 + k2) \ z^{(i+1)} &= zi + rac{1}{2}(k1 + k2) \
    \end{align}

Tabulated Results

  • Table provided to visualize the intermediate results leading to final computations.

  • Results show correlation between approximated values and computed errors using Heun’s method.

Error Calculation

  • Calculating errors helps in verifying the accuracy and reliability of the numerical approach:
    ext{Absolute relative true error} = rac{| ext{exact} - ext{approx} |}{| ext{exact} |} \times 100 \%

Conclusion

These examples illustrate the process of transforming higher order differential equations into a solvable first-order system, allowing you to model complex systems and arrive at approximate solutions effectively using numerical methods such as Euler's and Heun's. Each method has its own implications on accuracy and the rate of convergence towards the true solution, highlighting the importance of understanding these concepts in applied mathematics and engineering contexts.