PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Ordinary differential equations
Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise
Problem
The purely mathematical, dimensionless equation y′′′−3y′′+3y′−y=0 has initial data y(0)=1, y′(0)=0, y′′(0)=1.
((a)) Find the auxiliary polynomial and its roots, and write down the companion matrix A.
((b)) Determine the geometric multiplicity of the repeated eigenvalue of A (the eigenvector equation), decide whether A is diagonalisable, and explain how this is reflected in the form of the general solution of the equation.
((c)) Solve the initial-value problem for y(x).