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Hermite Differential Equation

The second-order ordinary differential equation

(d^2y)/(dx^2)-2x(dy)/(dx)+lambday=0.
(1)

This differential equation has an irregular singularity at infty. It can be solved using the series method

sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=1)^infty2na_nx^n+sum_(n=0)^inftylambdaa_nx^n=0
(2)
(2a_2+lambdaa_0)+sum_(n=1)^infty[(n+2)(n+1)a_(n+2)-2na_n+lambdaa_n]x^n=0.
(3)

Therefore,

a_2=-(lambdaa_0)/2
(4)

and

a_(n+2)=(2n-lambda)/((n+2)(n+1))a_n
(5)

for n=1, 2, .... Since (4) is just a special case of (5),

a_(n+2)=(2n-lambda)/((n+2)(n+1))a_n
(6)

for n=0, 1, ....

The linearly independent solutions are then

y_1=a_0[1-lambda/(2!)x^2-((4-lambda)lambda)/(4!)x^4-((8-lambda)(4-lambda)lambda)/(6!)x^6-...]
(7)
y_2=a_1[x+((2-lambda))/(3!)x^3+((6-lambda)(2-lambda))/(5!)x^5+...].
(8)

These can be done in closed form as

y=a_0_1F_1(-1/4lambda;1/2;x^2)+a_1x_1F_1(-1/4(lambda-2);3/2;x^2)
(9)
=a_0_1F_1(-1/4lambda;1/2;x^2)+a_2H_(lambda/2)(x),
(10)

where _1F_1(a;b;x) is a confluent hypergeometric function of the first kind and H_n(x) is a Hermite polynomial. In particular, for lambda=0, 2, 4, ..., the solutions can be written

y_(lambda=0)=a_0+1/2sqrt(pi)a_1erfi(x)
(11)
y_(lambda=2)=a_0[e^(x^2)-sqrt(pi)xerfi(x)]+xa_1
(12)
y_(lambda=4)=1/4{2e^(x^2)xa_1-(2x^2-1)[4a_0+sqrt(pi)a_1erfi(x)]},
(13)

where erfi(x) is the erfi function.

If lambda=0, then Hermite's differential equation becomes

y^('')-2xy^'=0,
(14)

which is of the form P_2(x)y^('')+P_1(x)y^'=0 and so has solution

y=c_1int(dx)/(exp(int(P_1)/(P_2)dx))+c_2
(15)
=c_1int(dx)/(expint(-2x)dx)+c_2
(16)
=c_1int(dx)/(e^(-x^2))+c_2=c_1erfi(x)+c_2.
(17)

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Cite this as:

Weisstein, Eric W. "Hermite Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HermiteDifferentialEquation.html

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