u-tt-u-xx-6x-0-x-lt-pi-t-gt-0-u-0-t-0-u-pi-t-pi-3-3pi-u-x-0-x-3-3x-3sin-x-u-t-x-0-0-

Question Number 100891 by bemath last updated on 29/Jun/20

u_{tt} = u_{xx} - 6 x; 0 ⩽ x < π, t > 0

u_{(0, t)} = 0; u_{(π, t)} = π^{3} + 3 π

u_{(x, 0)} = x^{3} + 3 x + 3 \sin x

u_{t} (x, 0) = 0

Answered by bramlex last updated on 29/Jun/20

u_{x x} - u_{t t} = 6 x \Rightarrow (D_{x}^{2} - D_{t}^{2}) u = 6 x

u = \frac{1}{D_{x}^{2}} (1 - \frac{D_{t}^{2}}{D_{x}^{2}}) (6 x)

\to T a y l o r s e r i e s {(1 - x)}^{- 1} = 1 + x + x^{2} + x^{3} + \dots

u = \frac{1}{D_{x}^{2}} (1 + \frac{D_{t}^{2}}{D_{x}^{2}} + \dots) (6 x)

u = \frac{1}{D_{x}^{2}} (6 x + 0 + \dots)

u = x^{3} + g_{1} (t) x + g_{2} (t)

u (0, t) & u (π, t) \to g_{1} (t) = 3, g_{2} (t) = 0

u_{1} (x, t) = x^{3} + 3 x

u_{x x} - u_{t t} = 0, s e t u = \emptyset (x + a t)

\emptyset^{″} (x + a t) \times a^{2} = 0 \to 1 - a^{2} = 0

a = \pm 1 \to u_{2} (x, t) = \emptyset (x + t) + \emptyset (x - t)

u = u_{1} + u_{2} ∵ u (x, t) = \emptyset (x + t) + \emptyset (x - t) + x^{3} + 3 x

s o u (x, 0) = x^{3} + 3 x + 3 \sin x

2 \emptyset (x) = 3 \sin x \to \emptyset (x \pm t) = \frac{3}{2} \sin (x \pm t)

u (x, t) = \frac{3}{2} \sin (x + t) + \frac{3}{2} \sin (x - t) + x^{3} + 3 x ★

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