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

2. Separation of Variables 2

3. Separation of Variables (cont.) 3

4. 4

5. Standard Form of Legendre’s Eq. 5

6. 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

7. General Form of Solution 7

8. Second Order Linear Differential Equations (SOLDEs) 8

9. Second Order Linear Differential Eqs. (SOLDEs) 9

10. Series Solutions – Ordinary Point 10

11. Series Solutions – Regular Singular Point 11

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

13. Regular Singular Point: Cases 2 and 3 13

14. Series Solutions – Irregular Singular Point 14

15. Classification of Singular Points - Example 15

16. Classification of Singular Points - Example 16

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