4. Assume that n > 2 and let +(x,t) be the fundamental solution of the heat equation in R" x (0, 0). Show that (r, t) satisfies the heat equation u = Au in R" x (0, o0).

Q: 4. The Forward-Difference method for solving ut = a²uxx is Uį,j+1 = (1 − 2λ)U¡‚j + λ[Ui+1,j +…

A: The forward difference formula for solving ut=α2uxx is given as Ui,j+1=(1-2λ)Ui,j+λ[Ui+1,j+Ui-1,j].…

Q: y" + 2y" + y = 5 sin t + colution to homogeneous part, the form of the parti olution is = C,cost +…

A: Solution of the differential equation

Q: Q:Solution of non-homogeneous Linear DE with constant coefficients? 3) (D'-5D?-2D+ 24)y x'e". Ans.:…

A:

Q: a2 (p 26. For the differential equation yt" + 2y" +y = 5 sint +7cost, given the follwoing solution…

A:

Q: 11. Show that U(x, y, z) = (x² + y² +z²) satisfies Laplace's partial differential equation + = 0.

A:

Q: Let y, = e*, y2 = xe* be two independent solutions of y" - 2y' + y = 0 ,x > 0 then the particular…

A: Option (A) is correct.

Q: Use the method of separation of variables and the superposition principle to solve the heat equation…

A:

Q: 3. Use your solution from Part 2 to give solutions to the IVPS i. y" + 2y' + 16y = cos 3t, y(0) =…

A:

Q: A guitar string of size L oscillates following the partial differential equation shown in the image.…

A:

Q: Write the general solution of the following Bessel equation xy`+xy'+(x² – 4)y=0

A: Given differential equation is, x2y''+xy'+x2-4y=0...1 The Bessel equation equation of a differential…

Q: By assuming that u(x, y,t) has the product form u(x, y, t) = 4(x+ Vt)ø(x – Vt)Y9y), separate the…

A:

Q: XP (A y dx + B x dy) + y (Cy dx + D x dy) = 0 where AD-BC # 0. xmyn,

A: Introduction: Integrating factor is related to first order ordinary differential equation. We…

Q: The function u depends on two independent variables: u= u(x, t). Find the general solution of the…

A: Given: The function u depends on two independent variables: u=u(x,t). PDE: ∂2u∂t2+∂u∂t+u=2t-x. To…

Q: What is the integrating factor of the linear De of order one y'= 1+ 3ytanx ? A B с D sec ³x cos³x *…

A:

Q: y=A x ln(x) + B x + F is the particular solution of the second-order linear DEQ: xy'' =8 where y'=2…

A: the solution of the given problem is below...

Q: If y1 and y2 two solutions of (1 – a²)y" – 2xy' + n(n+1)y = 0; -1<x < 1 Show that W (y1, Y2) : 1-…

A:

Q: Find all functions that form a fundamental set of solutions for t2y" – 2ty' + 2y = 0. O y(t) = et O…

A: We have to solve given problem:

Q: Solve the separable I.V.P y' (x2 +1) = x (y² + 4), y(0) = 0. y = 2 tan(2 + In(x + 1)) y = 2 tan(In(x…

A:

Q: Find the solution u (x,t) to the partial differential equation using the method of separation of…

A:

Q: Find all possible values of a, b, c so that U(x, t) = ax² + bt + ct² solves the heat equation Ut -…

A: To find the values of constants, we need to plug the given function into the differential equation.

Q: Assume that p, q : S' → R are continuous and p + 0. Find all solutions of the differential equation…

A: Given: Let us consider the given p,q:S1→R are continuous and ∫S1P≠0 and determine the al solutions…

Q: 24. Find all solutions of the one-dimensional heat equation that have the form w = e"t sin kx and…

A:

Q: Q3) Solve the heat equation u=u u t ¸ (0,1) = u(x,t) =0, t>0, u(x, 0) = 1, 00 with

A: Step 1:Step 2:Step 3:Step 4:

Q: use variation of parameters to find a general solution of the following differential equation=…

A: Given differential equation is y'''+3ty''=1t3. Also given the tree linearly independent solutions…

Q: Find the general solution of the inhomogeneous second-order linear ODE 2( 24" - 5' 3u)= 2(10 sinx)

A:

Q: Consider the partial differential equation 2² u du dx² Ət with boundary conditions u(0, ₺) = 0 and 1…

A:

Q: Verify that the given functions y1(t)=t, y2(t)=t?^t are solutions to the homogeneous equation…

A: To verify the  functions is solutions we derivative the functions then we find value in given D.E…

Q: Find the particular solution for the differential equation: y' + xe = -2x %3D for the boundary…

A: Topic = Differential equation

Q: Consider the differential equation 4y' = 3y + 2x/y Which of the following is the linear form of this…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: The following partial differential equation describes the diffusion of salt in a vessel. Assuming…

A: We are given the following system.                            ∂C∂t=D∂2C∂x2Initial condition: C(x,…

Q: For an infinitely long conducting cylinder of radius a, with its axis coincident with its z-axis,…

A: The voltage is u(r,θ) It satisfies the Laplace equation, ∇2u=0 --&gt;equation(1) Equation(1)…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.