Question

(1 point) Solve the boundary value problem by using the Laplace transform az w &w 16- dx2 x > 0, t> 0 at2 w(0,t) = 0, lim w(x, t) = 0, t> 0, X+0 w(x,0) = 2xe-*, dw F(x,0) = 0, x > 0, дt First take the Laplace transform of the partial differential equation. Let W be the Laplace transform of w. Then W satisfies the ordinary differential equation W" = subject to W(0) = and limxW(x) = Solve for W: W = Take the inverse Laplace transform, we then have the solution w(x, t) = (use step(x) for the unit step function of x).

Answer 1

dw at2 on care? Taking hoplace Transform s²w-sal(o) - W (0) = 1622 da² sw-sxoro = 162²W both side we have L(w). Dne 2 2²W/94) a sulleis - 16 an2 yaw s² = Ways) 16 s azw 22 St Willais) =0 auxillary equation m2 - 51 - be its m ille.at") 7 m=£5/4 fo n! son (695) = q e" + G / (5-ajn+) W(NU) = (008(,0)) = L( 2.66 ) = 2. !! (5+1) cs Scanned with CamScanner

lim Welfart = lin L(w (fy t)) my hapo, s) (font)) = o. maye - s/ym Ways) = G.ee the hilayo) = 9+G 22,= GAG Moyo Again hiffas as Gtg lin welcomes) -- o- Sm 7 ein (ees) ( MA faro 2 Car (5+1)2 cs you Hence, W(as) = (5+)? e "". (-5) = -2 -Elyon e 2 (St) Sam Will = 6 ) 4 TM (5+1) 2 52 (sti 2 8. W(0) = 2 (+1) ein wire) - ( ) -- CS anned with CamScanner

Date: & at me 2 e (5+12 2 -s/um W/4 3) = L(w/ry t)) (Stig? w/ryt) = l / 2 e 96) e S4112 Scanned with CamScanner