For n ≥ 1. Suppose u ∈ C^2(Rn ×[0,∞)) solves the heat equation u_t − ∆u = 0. Let u(x, t) = v((|x|^2)/t). Show that v satisfies 4*z*v′′(z) + (2n + z)*v′(z) = 0, where z = (|x|^2)/t. And find the general solution of the equation for v.

Q: Assume that all functions are continuous on some interval (a, b). Prove that if y₁ and y2 are…

A: Since we are given that y1 and y2 are solutions of the differential equations  y' + p(x)y = f1(x)…

Q: 20. Let z = (xo +x1)/2, h = (x, – xo)/2. To analyze the approximation (4.19), con- sider the error…

A: Given Let z = x0+x12, h =x1-x02 to analyze the approximation consider the error E =fx0,…

Q: Let n I be a positive integer. Show that the functional •1 S[y] = √² dx (y')"e", y(0) = 1, y(1) = A…

A: Given that, n>1 be a positive integer.

Q: 0, x<0, F(x) = { 공x?, O<x<2, 1, x22 a) Find E[X]. b) Find P(X 21|X)

A: From the distribution function we get:f(x)=ddxF(x)=ddx14x2=x2 for 0≤x&lt;2f(x)=0 otherwise. So,…

Q: A particle's position at time tis given by r(t)= (3t",4t+1,-t). Find a, att=3.

A:

Q: A. T(f(t)) = |B. T(ƒ(t)) = f(−t) from Pg to Pg = tª ƒ' (t) from P₂ to P5 = f(t)ƒ' (t) from P7 to P13…

A:

Q: 2. Evaluate aydA, where R = {(x, y) : 4 ≤ a² + y² ≤ 9, a ≤ 0, y ≥ 0}. x R

A:

Q: Find the partial derivative Əz Əx for the function z = 8y²√x.

A:

Q: = Let ƒ € Cm+¹ and the multiplicity of the root x* of f be n. In other words, f(x*) = ƒ'(x*) = =…

A:

Q: Let U ∼ Unif(0, 1). As a function of U, construct a r.v. X with desired CDFF(x) = 1 − e^−x^3 for x…

A:

Q: Q4. Simplify the following expression: 3-5 x 39 Click on image to enlarge if needed 12 81 243 324

A: To evaluate :- 3-5 * 39

Q: Use the Laplace transform to solve the system X +y -X = cost, x'+ 2y 0, x(0) = y(0) = 0. OA. x(t)…

A:

Q: 1. Let r = Vx2 + y² + z² and suppose f'(r) = df /dr exists. Prove that V{(r) = L("),

A:

Q: dt, then F'(x) = 0 a) If F(x) = √5 / / di 20 3 b) If F(x) = ² = dt, then F'(x) = c) If F(x) = √²/1/1…

A: Given:   a) Fx=∫20x1tdt      we use common integral ∫1tdt=lnt          =lnt20x          Fx=lnx-ln20…

Q: -a²k²1, Verify that the function u = heat conduction equation u, = α²uxx. = è¯ sin kx is a solution…

A:

Q: Find the linearization of f(x)=1−∫3x+124/(3+t)dt at x=2.   upper limit of integral = x+1 lower…

A: Linearization