HW: Solve the Partial Differential Equations by using separation of variables method:U1 – 4uxx = 0Uz(0, t) = uz(1, t) = 0u(x, 0) = f(x) = cos? nxU;(x, 0) = g(x)= sin² rx cos AX0 < x < 1, t > 0,t > 0,0

To solve the given partial differential equation by the method of saperation of variable we consider…

Answered: HW: Solve the Partial Differential…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider applying the method of separation of variables with u(x, t) = X(x) T(t) to the partial differential equation 82 u Əx² J² u Ət² du Ət Select the option that gives the resulting pair of ordinary differential equations (where is a non-zero separation constant). Select one: ○ X"(x) = µ, Ï(t) — Ï(t) = µ ○ X" (x) - X'(x) = μ₁ Ï(t) = µ µ₂ Ο X"(x) = µX(x), ○ X'(x) = µX(x), ○ X" (x) - X'(x) = µX(x), Ï(t) = µT(t) Ï(t) — Ï(t) = µT(t) T(t)-Ï(t) = µT(t)
Using the substitution y=vxanddy=vdx + xdv, the homogeneous differential equation [xsin ( - )-x]dx X A B C D |dx + xdy=0 reduces to (sinv-2v) dx + xdv=0 xsinvdx + vdy=0 sinvdx + xdv=0 (sinv + 2v) dx + xdv=0
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The differential equation can be written in differential form: where M(x, y) = y + 2y¹ = (y² + 6x) y' M(x,y) dx + N(x, y) dy = 0 and N(x, y) The term M(x, y) dx + N(x, y) dy becomes an exact differential if the left hand side above is divided by y7. Integrating that new equation, the solution of the differential equation is = = C.
Q5) Solve the partial differential equation V²u(x,y)=x with the condit u(x,0)=0 , u(x,10)=0, u(0,y)=0, u(20,y)=100 h=k=5
5) Suppose a path C is defined by r(t), 2 and r(8)= . If f is a differentiable function satisfying f(3,9) = 6 and f(6, 10) = 14, what is the value of Vf dr?
Please use PDE(partial differential equation) content to solve this problem, thank you. The second picture shows the (1) and (2). The question is asking to use (1) and (2) to prove the problem 2 is true.
How to solue X-2 by iteration? %3D X+2

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HW: Solve the Partial Differential Equations by using separation of variables method: Ut - 4uxx = 0 Uz(0, t) = uz(1, t) = 0 u(x, 0) = f(x) = cos² t x u;(x, 0) = g(x)= sin² n x coS T X 0 < x < 1, t > 0, t> 0, 0