y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant, is...

y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant, is known as the Hermite equation, named after the famous mathematician Charles Hermite. This equation is an important equation in mathematical physics.

• Find the ﬁrst four terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions

• Observe that if λ is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for λ = 0,2,4,6,8,10. Note that each polynomial is determined only up to a multiplicative constant.

• The Hermite polynomial, Hn(x), is deﬁned as the polynomial solution of the Hermite equation with λ = 2n for which the coeﬃcient of xn is 2n. Find H0(x),...,H5(x).

• Give some plots of these polynomials

Expert Solution

Colby Messinger answered 1 year ago

Consider the ODE y''(x) = λy(x) for some real constant λ. Determine ALL values of λ...

Consider the ODE y''(x) = λy(x) for some real constant λ. Determine ALL values of λ for which there exists solutions satisfying the boundary conditions y(0) = y(10) = 0. For each such λ, give all possible solutions. Are they unique?

Find the eigenvalues of the problem: y'' + λy = 0, 0 < x < 2π,...

Find the eigenvalues of the problem: y'' + λy = 0, 0 < x < 2π, y(0) = y(2π), y' (0) = y'(2π) and one eigenfunction for each eigenvalue.

x^2 y ′′ + xy′ + λy = 0 with y(1) = y(2) and y ′...

x^2 y ′′ + xy′ + λy = 0 with y(1) = y(2) and y ′ (1) = y ′ (2) Please find the eigenvalues and eigenfunctions and eigenfunction expansion of f(x) = 6.

x^2y''+2xy'+x^2y=0 Determine y if y1=(sin(x))/x

Solve the differential equation 1. a) 2xy"+ y' + y = 0 b) (x-1)y'' + 3y...

Solve the differential equation 1. a) 2xy"+ y' + y = 0 b) (x-1)y'' + 3y = 0

Solve the differential equation y''(x)-2xy'(x)+2ny(x)=0 using the Hermite Polynomials

Let Q1=y(2) Q2 =y(3) where y=y(x) solves y'+2xy=2x^3 y(0)=1 dy/dx= (2x-y)/(x+4y) y(1)=1 y'+ycotx=y^3sin^3x y(pi/2)=1 (y^2-2x)dx+2xy dy=0 ...

Let Q1=y(2) Q2 =y(3) where y=y(x) solves y'+2xy=2x^3 y(0)=1 dy/dx= (2x-y)/(x+4y) y(1)=1 y'+ycotx=y^3sin^3x y(pi/2)=1 (y^2-2x)dx+2xy dy=0 y(1)=1 2x^2y;+4xy=3sinx y(2pi)=0 y"+2y^-1 (y')^2=y'

Solve the initial value problem. y'=(y^2)+(2xy)+(x^2)-(1), y(0)=1

Find the eigenvalues and eigenfunctions of the problem y" + λy = 0 , y(−π/2) =...

Find the eigenvalues and eigenfunctions of the problem y" + λy = 0 , y(−π/2) = 0 , y(π/2) = 0. Please explain in detail all steps. Thank you

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x <...

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y < infinity with parameters λ > 0 and μ > 0 a) Find c such that this is a PDF. b) Show that X and Y are Independent c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in exponential form d) Find the marginal distribution of Y, f(y) e) Find E(Y)

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