1. D. R. Wilton ECE Dept. ECE 6382 Second Order Linear Differential Equations

2. Separation of Variables

3. Separation of Variables (cont.)

5. Standard Form of Legendre’s Eq.

6. Separation of Variables 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

7. General Form of Solution

8. Second Order Linear Differential Equations (SOLDE)

9. Second Order Linear Differential Eqs. (SOLDEs)

10. Series Solutions – Ordinary Point

11. Series Solutions – Regular Singular Point

12. Series Solutions – Irregular Singular Point x = Re z Im z R a nearest singularity

13. Classification of Singular Points - Example

14. (2) (2) (3) (3) (4) (4) Classification of the Point at Infinity

15. The End