Find uxt that solves the following problem Your solution may

Find u(x,t) that solves the following problem: Your solution may contain unevaluated infinite sums, but not unevaluated integrals.

Data.

ut = ux,x + 1

u(0,t) = u (,t) = 0

u(x,0) = x(x-)

Answer.

Solve for:

ut - ux,x = 0

let u be;

u = e^ht^+kx

he^ht^+kx - k²e^ht^+kx = 0

h = k²

replacing we have;

c.f = e^k2t^+kx

now,

P.I = 1/D´ - D²

= 1/D´[1- D²/D´]^-1

= 1/D´(1+0+..+n) = t

so,

u(x,t) = e^k2t^+kx + t

u(x,0) = x(x-)

e^kx = x(x-)

kx = log[x(x-)]

u(0,t) = 0

0 = e^k2t^+kx + t

0 = e^k2t x e^kx + t

= -t[x(x-)]+t

= x - tx² + t

u(x,t) = x - t(x² - 1)