Solve the partial differential equation:a²uazu= c²by the method of separation of variables for a vibrating string subjected to31²8x2¹boundary conditions:Xoineu(0, t) = 0u(l, t) = 0And initial conditions:1407-4x² -5ineu(x,0) = u, sinди-(x,0) = 0at(7)In LCimeMe

Given: ∂2u∂t2=c2∂2u∂x2 To find: ux,t=0

Answered: 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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Find the solution. Zy" - 3y' + y = 0 y = c,e/?+c,e' y = c,e/+c,e* B
Show the complete solution, choices are in the second image.
Use the substitution y(x)= x* . u(x)to convert the equation d'y dy +ax- dx? dx (1+ **x* ) y = 0 x? in which ais a parameter, into an equation for "(x).Find the values of a and Athat make the equation in ux()Bessel's equation of order zero. Use the result to find the general их solution yx() that corresponds to the value of a.
3t --- 3t a. Verify that y(t) = b. Verify that yo(t) = = is a solution to y '(t) substituting it into the differential equation. (Enter the terms in the order given.) y' (t) = 5t + 1 2t 4t + 2 = Use angle brackets, ( and ), to indicate vectors, and separate the components with commas. E.g. (e', t², 2e²¹). = 1 -2 2 -2 2 1 2 2 1 1 [ -2 is a particular solution to -2 21 1 2 1 2 y(t) + 2 y(t) by [1 4- (4t+ 4+ 5t) 0 2 (14t+2+ 4t) by substituting it into the differential equation. (Enter the terms in the order given.) Use angle brackets, ( and ), to indicate vectors, and separate the components with commas. E.g. (3, 3t², 2t + 1).
Find yc and yp (second oreder)
3. A projectile is fired with an initial velocity of 128 ft/s from a cannon at time t = 0 from a height of 32 ft above the ground and at an angle of 30° with the horizontal. (a) Determine: x '(0) and y'(0). (These values represent the horizontal and vertical components of the initial velocity vector.) (b) Use integration to determine the position equations: x(t) and y(t) for the projectile t seconds after being fired by assuming that the projectile is in freefall. Thus, x"(t)=0 ft/s² and y"(t)=-32 ft/s² for all t≥ 0. (Note: x"(t) and y"(t) are the horizontal & vertical components of the projectile's acceleration vector at time t.) (c) Determine the maximum height, H, of the projectile and its horizontal distance, D, from the launch site at the time that it achieves this height. (Hint: The maximal height is attained at the time, t*, at which y'(t*) = 0.) (d) Determine the range of the projectile, R. YA (0,32) - P. 32' 128 0=30° x'(0) y'(0) H R Pr=(x(t), y(t)) is the position of the…
Using the substitution y=vxanddy=vdx + xdv, solve the homogeneous differential equation y [xsin (--)-> ]dx- X y dx + xdy=0 + y y @ x (exe ² - - co₁²-) = c A CSC- X X y y ℗ x(exe²= - co²=-=) = c B y csc X X y y Ⓒy[ese ² + + co₁² ) = c C X X © x (exe ² + + co₁=² ) = c y y D X X
Solve for the DE of: y = C₁ + С₂e7x O y'-+7y=0 Oy" - 7y¹ = 0 O y' - 7y=0 Oy" + 7y' = 0
please do b) only

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Solve the partial differential equation: a²u azu by the method of separation of variables for a vibrating string subjected to boundary conditions: 8x2¹ ine u(0, t) = 0 . u(l, t) = 0 And initial conditions: A=0 u(x, 0) = u, sin ди InuYL -(x,0) = 0 (x, at πχ Me -51ME