Hermite Differential Equation and Hermite Polynomials 

Yo bro Im here for this reason ...QM

Good catch, and thanks for clarifying! I'll put an edit in the description accordingly!

Exponentital generating function for Hermite polynomials is easier to get starting from recurrence relation H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x) H_{0}(x) = 1 H_{1}(x) = 2x H_{n+2}(x)=2xH_{n+1}(x)-2(n+1)H_{n}(x) H_{0}(x) = 1 H_{1}(x) = 2x E(x,t) = sum(H_{n}(x)*t^n/n!,n=0..infinity) After some calculations you will get following initial value problem B''(t) + 2(t -x)B'(t)+2B(t)=0 B(0) = 1 B'(0) = 2x Particular solution to this equation is easier to guess after reduction to Riccati Reduction to Riccati y'' + p(x)y'+q(x)y = 0 y'' = - p(x)y' - q(x)y | :y y''/y = -p(x)*(y'/y) - q(x) | -(y'/y)^2 y''/y - (y'/y)^2 = -(y'/y)^2 - p(x)*(y'/y) - q(x) y''/y - y'*y'/y^2 = -(y'/y)^2 - p(x)*(y'/y) - q(x) (y''*y-y'*y')/y^2 = -(y'/y)^2 - p(x)*(y'/y) - q(x) (y'/y)' = -(y'/y)^2 - p(x)*(y'/y) - q(x) Let y'/y = z z' = -z^2 -p(x)z - q(x) y' = yz Reduction Riccati to Bernoulli assuming you somehow guessed particular solution Let z_{1} be the particular solution to Riccati equation z' = p(x)z^2 + q(x)z + r(x) (1) z_{1}' = p(x)z_{1}^2 + q(x)z_{1} + r(x) (2) Lets subtract eq no 2 from eq no 1 z' - z_{1}' = p(x)(z^2-z_{1}^2)+q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})(z + z_{1}) + q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})(z - z_{1} + 2z_{1}) + q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})(z - z_{1}) + 2z_{1}p(x)(z-z_{1}) + q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})^2 + (2z_{1}p(x) + q(x))(z-z_{1}) (z-z_{1})' - (2z_{1}p(x) + q(x))(z-z_{1}) = p(x)(z - z_{1})^2 w = z-z_{1} w' - (2z_{1}p(x) + q(x))w = p(x)w^2

He said in the beginning of the video: quantum mechanical harmonic oscillator. For details of the derivation, see: opentextbc.ca/universityphysicsv3openstax/chapter/the-quantum-harmonic-oscillator/ On the webpage above, search for keywords “time-independent Schrödinger equation”. Hope it helps.

That a good idea!

I am making videoes on that this week. Maybe u can check my channel up by then. Thanks

M planning to do hermite polynomial, airy's, laguerre, laplacian and mathieu....the whole shabang

Did you try turning them off??