smashmath

math phd student, dork extraordinaire.

i like differential equations and linear algebra.

here are some math posts ive written if you want to look at them i guess

if you want to support me, i have a ko-fi.

personal favorites:

• Constant Coefficient ODEs Made Simple with Linear Operators
• Shortcuts for Finding Eigenvalues and Eigenvectors
• Solving systems of first-order ODEs like a baller
• The Alpha Method (Generalized Exponential Response Formula)

my research stuff:

things ive discovered independently derived. i think they’re all cool, but only a few of them are actually useful, in my opinion.

• Function Interpolation: a method to get a function (which is a linear combination of some given set of basis functions) that satisfies certain conditions using determinants, given that one exists and is unique. for example, a determinant which gives the unique lowest degree polynomial that passes through a certain set of points.

• A formula for some particular solutions to certain ODEs: a determinant formula which gives a particular solution to any linear constant-coefficient ordinary differential equation which has a forcing function of exponential nature (ex. \(g(t)=t^ne^{\alpha t}\cos(\beta t)\)). Uses results from Function Interpolation.

• Constructing integer systems of differential equations with integer solutions: methods to construct nice systems with nice solutions. useful for professors/textbook authors to make good lecture examples or exam problems. somewhat of a work in progress. i also have a post for second order systems.

• matrix exponential stuff: i really love matrix exponentials.
  1. Matrix Exponential Formulas for 2x2 Matrices
  2. Matrix Exponentials Using Differential Equations
  3. Exponentials of Symmetric Matrices Using the Spectral Theorem
  4. Matrix Exponential Formulas for 2x2 Matrices Using Laplace Transforms
  5. Another approach to matrix exponential formulas: coming soon…
• New Ways to Calculate Normalized Solutions to Linear Constant-Coefficient Differential Equations: solve just one set of \(n\) first-order initial value problems to get the \(n\) normalized solutions to an \(n\)-th order differential equation. this should be the fastest way to find them using a computer. alternatively, find one normalized solution and get the others recursively.