Introduction to differential equation

• $$ \frac{dy}{dx}=\frac{-y}{x}$$

  Is ordinary differential equation because it con...}

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Hermite Differential Equation

• The form of Hermite differential equation

  $$y''-2xy'+2ny=0\dotsm(1)$$

...}
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Series solution and Hermite polynomial of Hermite differential equation

• $$y=\sum_{J=0}^N\frac{(-1)^J n! a_n x^{n-2J}}{2^{2J} J! (n-2J)!}$$

  Where N= n/2...}

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Generating function and Rodrigue's formula of Hermite polynomial

Important equation to be remember

• $$e^{2xt-t^2}=\sum_{n=0}^\infty\frac{H_n(x) +n}{n!...}
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Orthogonality property of Hermite polynomial

These are the important equation to be remember in this chapter

• $$\frac{d^2y}{dx^2}-...}
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Bessel differential equation and it's series solution

You must remember these equation

• \(J_{-n}(x)= (-1)^n J_n(x)\)
• $$ J_{-n}(x)=...}
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Generating function of Bessel polynomial and Recurrence relation

Important equation to be remember

• The generating function of Bessel polynomial i...}

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Integral form of Bessel polynomial and orthogonality of Bessel differential equation

Equation to be remember:

• $$\int_0^p rJ_n(ar) J_n(br) dr=0$$

  This is the o...}

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Series solution of Laguerre's Differential Eqution

Important equation to be remember:

• $$y=e^{\frac{-x}{2}} x^{\frac{(p-1}{2}} L_n^k...}

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Generating function, Recurrence relation and Rodrigues formula of Laguerre's Differential Equation

Equations to be remember:

• The generating function of Laguerre polynomial is

  ...}
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First four Lagueree polynomial, Orthogonality of Lagueree polynomial

Important equation to be remember:

• The Rodrigue's formula for Lagueree polyomial...}

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Series solution of Legendre differential equation

Important equation to be remember:

• $$(1-x^2) \frac{d^2y}{dx^2} - 2x \frac{dy}{dx...}

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Polynomial solution,Generating function of Legendre differential equation

Important equations to be remember

• The generating function fo Legendre different...}

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Recurrence relation and Rodrigue's formula of Legendre differential equation

Important equat...}

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Orthogonality of Legendre differential equation and Associate Legendre differential equation

Important equation to be remember

• $$(1-x^2)\frac{d^2y}{dx^2}-2x \frac{dy}{dx}+n(n+1)...}
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Co-ordinate system and it's rotation.

1.  $$\sum_{p=1}^3 l_{pm}l_{pn}=1 if m=n$$

   $$\sum_{p=1}^3 l_{pm}l_{pn}=0 if m\...}

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