6. Let u(x, t) denote the solution of the following initial-boundary value problem u du 0 < x < 1, t> 0 20t Uz(0, t) Uz(1, t) = 0 u(x, 0) What is the behaviour of the solution when t approaches to oo? D) 2 1 A) o B) -00 C) 0 E) 2

Q: 6. Find the absolute maximum and minimum of the function f (x, y, 2) = x + 2y on the intersection…

A:

Q: 3.) Use the Maximum/Minimum Principles to deduce that the solution u of the problem D.E. ut = kuzr…

A: given differential equation ut=kuxx    ...i where 0≤x≤π, t≥0 boundary condition u0,t=0       uπ,t=0…

Q: dy dx + (-²) ₁ y = (x³ - 3)y² x.

A:

Q: 5. Solve V'u = 0 0<r<1 [1 u(1,4) 0 < 0 < T/2 T/2 < 6 < T -1 This is the problem where the boundary…

A: Given: ∇2u = 0, 0&lt;r&lt;1 &amp; u(1,ϕ) = f(ϕ) (let) = 10≤ϕ≤π2-1π2&lt;ϕ≤π

Q: Find the functions x*(r) and y*(r) such that x = x*(r) and y = y*(r) solve the problem max f(x, y,…

A: max fx , y, r=max-x2-xy-2y2+2rx+2ry To Find: x*r , y*r such that x=x*r ,y= y*r.

Q: 1. Solve U = Uzz, 00 subject to boundary conditions and initial conditions u(0,1)=u₂(2. t) = 0…

A:

Q: Use the Maximum Principle to solve the following dynamic optimization problem 2 (1+ *¢ – u?), X+1 =…

A: We are given a dynamic optimization problem and we have to solve for the optimal solution using…

Q: 2. Solve the following Cauchy problem: u = u -0 0, u(x, 0) = 1 -1, u(1,0) = 4r, -00 <I< 00.

A:

Q: Find a particular solution to y"+2y' + y =t

A:

Q: Find the solutions x*(r, s) and y* (r, s) of the problem max f(x, y, r, s) = max(r²x + 3s²y = x² –…

A: max fx , y , r,s=maxr2x+3s2y-x2-8y2 To Find: x*r , s , y*r , s

Q: 23. Solve the Cauchy problem 1 (r+ y) uz + (r – y) uy = 1, u (1, y) = %3D

A: Given: x+yux+x-yuy=1 , u1,y=12

Q: 3. Show that the initial value problem * = |x|¹/² x(0) = 0 has four different solutions through the…

A: Given: Let us consider the given IVP, x˙=x12x0=0 To determine: Graph these solutions in the t, x…

Q: where co= 1. Enter T= s) Find the solution of z²y" + 5ay + (4-4x)y=0, z> 0 of the form Cn= , n = 1,…

A: Given a differential equation and y1 is it's solution. ..then we doing derivative  of y1 and putting…

Q: Solve Ut = Urx, x > 0, t> 0, u (x, 0) = f (x), u (0, t) = 0.

A: Given partial differential equation isut=uxx,     x&gt;0,     t&gt;0ux, 0=fxu0, t=0

Q: Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation Ut + UU x =…

A: The given partial differential equation is .The initial value is .We know that the characteristic…

Q: Consider y" + 4y' + 5y = ae -3* 8 (t - 31), y (0) = 1, y (0) = -2. Зл Find the value of a such that…

A:

Q: . Prove the stability estimate for the solutions of the boundary-value problem -(au)' + bu' + cu =…

A: As per the question, we are asked to prove the stability estimate for the solutions of the…

Q: Solve the following IBVP: Utt = 4uzz 0 0, u, (0, t) = u,(27, t) = 0, t>0, u(x,0) = 0, 0<I< 2n, T.…

A:

Q: Solve Laplace's equation, u(0, y) = 0, u(x, 0) = f(x), u(x, y) = a²u əx² 8²u 0, 0<x<a, 0 <y<b, (see…

A:

Q: If (y1.V2) is a fundamental set of solutions for the DE: t'y" -3y'+costy = 0 and if W(v1,V2)(1) = 2,…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 4. Assume y(x) : = ea" cosBx is one of the solutions of the DE у3) — Зу" + 7у - 5у —D 0. Then a + B…

A:

Q: If x = t2- 2t+1 and y=t²-2t then stationary point is O None of these O (-4,-1) O (0,1) O (0, -1) О…

A:

Q: . (a) Solve the boundary value problem Ju + Be-12 at where u(0, t) = 0, u(L, t) = 0, u(x,0) = f(x),…

A: As per the question we are given an initial boundary value problem which we have to solve for u(x,t)…

Q: )Please answer in legible and clear handwriting.

A: Given that  xn,xn-1,....,xn-i be a sequence of successive approximation vectors to the solution of…

Q: х+ 2y - z %3D 1 = 1 X - = 2 y + z = 3

A: Let   x+2y-z=1↦Equation 1x-y=2↦Equation 2y+z=3↦Equation 3 by taking Eq 1-Eq 2, we get…

Q: 3. Let u(x, t) be the solution to 5ætu, – ut = 0, u(x,0) = 3x², x > 0 - obtained using the method of…

A:

Q: 33 prone that ucx, y) = x F(2x+y) is a general %3D Solution of x Ux-2xUy = Ucxyy) then find the…

A:

Q: 2. Consider the equation with the initial condition удҳи — и²дуu = 4x u(x,x²) = -2x for any x E R.…

A:

Q: 5. (a) Solve the cauchy problem given by x²ux + xyuy + xu = Xx-y where (x, y) + (0,0) if u(x, xln x)…

A: Given x2ux+xyuy+xu=x-y where x,y≠0,0 if ux,xlnx=-lnx, x&gt;0

Q: Solve the problem: Ut - 2 Uxx u (x, 0) u (0, t) = = h, = x, sint, t> 0, 0 < x < 1, 0 ≤ x ≤ 1, ur (1,…

A: The given PDE is:ut−2uxx=hfor 0&lt;x&lt;1,t&gt;0with the following initial and boundary…

Q: 3) Find the solution of initial and boundary value problem Ut - 2Uxx = 0, 0 0 u (0. t) u (π, t) = 0,…

A:

Q: 2. Show that if u is a SC

A: Given, if u is a solution of                          ∇2u+D(x, y)∂u∂x+E(x, y)∂u∂x=-F(x, y) in D with…

Q: Let u(x, t) be solution of Utt - Uxx = Xxt -00 0 u(x, 0) = u(x, 0) = 0, Then u(2, 3) is equal to (a)…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.