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ECE 6382. Second Order Linear Differential Equations. D. R. Wilton ECE Dept. Separation of Variables. Separation of Variables (cont.). Separation of Variables (cont.). Standard Form of Legendre’s Eq.
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ECE 6382 Second Order Linear Differential Equations D. R. Wilton ECE Dept.
Most equations of mathematical physics are linear(ized) second order partial differential equations: • Wave equation • Heat equation • Navier-Stokes equation • Dirac equation • As above, if applicable, the separation of variables method leads to second order linear differential eqs. (SOLDEs) • Harmonic eq. • Bessel’s eq. (cylindrical and spherical) • Jacobi, Chebyshev, Legendre, Laguerre, Hermite eqs. • Laplace’s, Poisson’s equation • Klein-Gordon equation • Schrödinger equation Separation of Variables
nearest singularity Im z x = Re z a R Series Solutions – Irregular Singular Point