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Full-Text Articles in Science and Mathematics Education
Modeling Aircraft Takeoffs, Catherine Cavagnaro
Modeling Aircraft Takeoffs, Catherine Cavagnaro
CODEE Journal
Real-world applications can demonstrate how mathematical models describe and provide insight into familiar physical systems. In this paper, we apply techniques from a first-semester differential equations course that shed light on a problem from aviation. In particular, we construct several differential equations that model the distance that an aircraft requires to become airborne. A popular thumb rule that pilots have used for decades appears to emanate from one of these models. We will see that this rule does not follow from a representative model and suggest a better method of ensuring safety during takeoff. Aircraft safety is definitely a matter …
A Generalized Solution Method To Undamped Constant-Coefficient Second-Order Odes Using Laplace Transforms And Fourier Series, Laurie A. Florio, Ryan D. Hanc
A Generalized Solution Method To Undamped Constant-Coefficient Second-Order Odes Using Laplace Transforms And Fourier Series, Laurie A. Florio, Ryan D. Hanc
CODEE Journal
A generalized method for solving an undamped second order, linear ordinary differential equation with constant coefficients is presented where the non-homogeneous term of the differential equation is represented by Fourier series and a solution is found through Laplace transforms. This method makes use of a particular partial fraction expansion form for finding the inverse Laplace transform. If a non-homogeneous function meets certain criteria for a Fourier series representation, then this technique can be used as a more automated means to solve the differential equation as transforms for specific functions need not be determined. The combined use of the Fourier series …
Undetermined Coefficients With Hyperbolic Sines And Cosines, Laurie A. Florio, George L. Fischer
Undetermined Coefficients With Hyperbolic Sines And Cosines, Laurie A. Florio, George L. Fischer
CODEE Journal
The method of undetermined coefficients is commonly applied to solve linear, constant coefficient, non-homogeneous ordinary differential equations when the forcing function is from a selected class of functions. Often the hyperbolic sine and cosine functions are not explicitly included in this list of functions. Through a set of guided examples, this work argues that the hyperbolic sine and cosine ought to be included in the select class of functions. Careful explanation is provided for the necessary treatment of the cases where the argument of the hyperbolic sine and/or cosine functions matches one or both of the roots of the characteristic …
Special Case Of Partial Fraction Expansion With Laplace Transform Application, Laurie A. Florio, Ryan D. Hanc
Special Case Of Partial Fraction Expansion With Laplace Transform Application, Laurie A. Florio, Ryan D. Hanc
CODEE Journal
Partial fraction expansion is often used with the Laplace Transforms to formulate algebraic expressions for which the inverse Laplace Transform can be easily found. This paper demonstrates a special case for which a linear, constant coefficient, second order ordinary differential equation with no damping term and a harmonic function non-homogeneous term leads to a simplified partial fraction expansion due to the decoupling of the partial fraction expansion coefficients of s and the constant coefficients. Recognizing this special form can allow for quicker calculations and automation of the solution to the differential equation form which is commonly used to model physical …