Second Order Linear Differential Equations ECE 6382 Notes are from D. R. Wilton, Dept. of ECE David R. Jackson 1

Separation of Variables 2

Separation of Variables (cont.) 3

4

Standard Form of Legendre’s Eq. 5

Separation of Variables  Most equations of mathematical physics are linear 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 equations  Klein-Gordon equation  Schrödinger equation 6

General Form of Solution 7

Second Order Linear Differential Equations (SOLDEs) 8

Second Order Linear Differential Eqs. (SOLDEs) 9

Series Solutions – Ordinary Point 10

Series Solutions – Regular Singular Point 11

Series Solutions – Regular Singular Point x y R a Nearest singularity x  z 12

Regular Singular Point: Cases 2 and 3 13

Series Solutions – Irregular Singular Point 14

Classification of Singular Points - Example 15

Classification of Singular Points - Example 16

(2) (2) Classification of the Point at Infinity 17