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D. R. Wilton ECE Dept.

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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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D. R. Wilton ECE Dept.

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

  2. Separation of Variables

  3. Separation of Variables (cont.)

  4. Separation of Variables (cont.)

  5. Standard Form of Legendre’s Eq.

  6. 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

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

  13. Classification of Singular Points - Example

  14. Classification of the Point at Infinity

  15. The End

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