In Problems 1–10 solve Laplace’s equation (1) for a rectangular plate subject to the given boundary conditions.
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Differential Equations with Boundary-Value Problems (MindTap Course List)
- Ex. 5. Find a solution of Laplace's equation, u +u„ =0, inside a rectangle subject to the following boundary conditions: а. и(0, у) 3 0, и(, у) 3 0, и(х,0) — -4sin (2rx). и(х,5) %3 6sin (3rx). b. и(0), у) - 0, и(, у) - 0, и(х,0) —х', и(x,2) -0. с. и, (0, у) 3 5sin (ту). и,(1, у)-13sin (2тy), и(х,0) — 0, и(х,2)-0. d. u, (0, у) — 0, и(1, у) — 0, и(х,0) —0, и, (х, 2) %35 сos 2 е. и, (0, у) - 0, и, (2л, у) - 0, и(х,-1)-0, и(х,) —1+sin (2xх).arrow_forwardWhich of the following is NOT a possible solution for Laplace's equation? (a) y = (AePx + Be-P*)(Ccos py + Dsin py) (b) y = (Acos px + Bsin px)(CEPY + De PY) (c) y = (Ax + B)(Cy + D) (d) y = (A P* + Be-P*)(CePy + Depy) O a O b O carrow_forward1. Find all singular and all ordinary points of the DE (0.5x²+2x+25)y"+xy'+y=0arrow_forward
- The equations ry² + 6xzcos(u) + ye" = x'yz + xe" – 7u?v² = 21 are solved for u and v as functions of x, y and z near the point P where (x,y,z)=(1,1,1) and (u, v) = (5,0). Find ()zy at P. 12 and %3| 12,00 -70,00 -30,00 -0,17 56,00arrow_forward1. Solve Laplace's equation inside a rectangle 0 0 and H > 0), with the following boundary conditions: a) u(0, y) = g(y), u(L, y) = 0, (r, 0) = 0, and u(x, H) = 0. b) u(0, y) = g(y), u(L, y) = 0, (x,0) = 0, and (x, H) = 0.arrow_forward5. The function r(t) moving in 3D space. = (2 sin (¹), 3, 2 cos (t)) parametrizes (draws) the path of a particle (a) Evaluate r(t) at t = 0, 1, 2, 3, 4.arrow_forward
- 4. Determine when the following pairs of functions are linearly independent. (a) yı(t) = erit; y(t) = er²t, r₁,72 € R (b) y(t) = cos(at); 32(t) = sin(at), a = R (c) y₁ (t) = cosh(at); y₂(t) = sinh(at), a € Rarrow_forward3. Solve the Laplace equation inside a 60° wedge of radius 1 subjected to the boundary conditions du (г.0) 3 0 r,-|=0, 3 u(1,0) = sin(20) du r,-|=0 3 u(1,0) = sin(20) (r,0)=0 +arrow_forward15 Find ( T + In y) dydxarrow_forward
- 4. Verify that the functions 1, r, cos r and sinr form a funda- mental set of solutions of y(4) + y" = 0 on (-0o, 00).arrow_forwardProblem 3. Prove that F x = y Ay = z → x = z.arrow_forward- 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(e) and y = usin(0) + v cos(0).arrow_forward
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