Preface
This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.
Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in normal font. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.
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Reduction Higher Order ODEs
Sometimes higher order differential equations can be reduced to lower and first order equations. We consider two classes of equations when this is possible.
Dependent Variable Missing
For a differential equation of the form \( y^{(n)} = f(x, y^{(n-1)} ) , \) the substitution \( p = y^{(n-1)} , \ p' = y^{(n)} \) leads to a first order equation of the form \( p' = f(x,p) . \) If this equation can be solved for p, then y can be obtained by integrating \( {\text d}^{n-1} y/{\text d}x^{n-1} = p . \) Note that one arbitrary constant is obtained in solving the first order equation for p, and n-1 constants are introduced in the integration for y.
Example: Consider the differential equation with dependent variable missing:
Independent Variable Missing
Consider second order differential equation of the form \( y'' = f(y,y') , \) in which the independent variable t does not appear explicitly. If we let \( v = y' = \dot{y} , \) then we obtain \( {\text d} v/{\text d}t = f(y,v) . \) Since the right side of this equation depends on y and v, rather than on t and v, this equation contains too many variables. However, if we think of y as the independent variable, then by chain rule, \( {\text d} v/{\text d}t = \left( {\text d} v/{\text d}y \right) \left( {\text d} y/{\text d}t \right) = v \left( {\text d} v/{\text d}y \right) . \) Hence the original differential equation can be written as \( v \left( {\text d} v/{\text d}y \right) = f(y,v). \) Provided that this first order equation can be solved, we obtain v as a function of y. A relation between y and t results from solving \( {\text d} y/{\text d}t = v(y) , \) which is a separable equation. Again, there are two arbitrary constants in the final answer.
Example: Consider the differential equation \( y'' + y \left( y' \right)^3 =0 . \) Upon setting \( y' =v , \) we reduce the given differential equation of the second order to another one: \( v \, \frac{{\text d}v}{{\text d}y} + y \left( v \right)^3 =0 . \) Assuming that \( v \ne 0 , \) we reduce it to the first order equation
Example: Dr. D.H. Parsons derived in 1963
- D.H. Parsons, Exact Integration of the Space-Charge Equation for a Plane Diode: A Simplified Theory, Nature, Volume 200, October, 126--127, 1963.
The above second order differential equation admits two integrating factors, one less obvious than the other:
In many practical problems, a second order differential equation can be reduced to an exact equation. For example, consider the equation of motion for the damped harmonic oscillator:
- K. Bohlin, ntegrationsmethode f ̈ur lineare Differential-gleichungen, Astron. Iakttag. Undersokn. Stockholms Observ., Volume 9, 3--6, 1908.
The equation of motion for the damped harmonic oscillator may be derived from the Lagrangian
Let us simplify the Hamiltonian with the transformation
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