Question

2. a. Show that the general solution of the PDE $u_{xy} = 0$ is $u(x, y) = F(x) + G(y)$ for arbitrary differentiable functions $F$, $G$. b. Use change of variable $\xi = x + t$, $\eta = x - t$ to show that $u_{tt} - u_{xx} = 0 \iff u_{\xi\eta} = 0$. c. Use a. and b. to rederive d'Alembert's formula.

          2. a. Show that the general solution of the PDE $u_{xy} = 0$ is
$u(x, y) = F(x) + G(y)$
for arbitrary differentiable functions $F$, $G$.
b. Use change of variable $\xi = x + t$, $\eta = x - t$ to show that
$u_{tt} - u_{xx} = 0 \iff u_{\xi\eta} = 0$.
c. Use a. and b. to rederive d'Alembert's formula.

2. a. Show that the general solution of the PDE uxy = 0 is
u(x, y) = F(x) + G(y)
for arbitrary differentiable functions F, G.
b. Use change of variable ξ = x + t, η = x - t to show that
utt - uxx = 0 uξη = 0.
c. Use a. and b. to rederive d'Alembert's formula.

Added by Crystal B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

10/08/2023

Step 1

We are given the PDE $u_{xy} = 0$. We integrate this equation with respect to $x$ to get $\int u_{xy} dx = \int 0 dx$ $u_y = F(y)$ where $F(y)$ is an arbitrary function of $y$. Now we integrate with respect to $y$ to get $\int u_y dy = \int F(y) dy$ $u(x, y) = Show more…

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2. a. Show that the general solution of the PDE $u_{xy} = 0$ is $u(x, y) = F(x) + G(y)$ for arbitrary differentiable functions $F$, $G$. b. Use change of variable $\xi = x + t$, $\eta = x - t$ to show that $u_{tt} - u_{xx} = 0 \iff u_{\xi\eta} = 0$. c. Use a. and b. to rederive d'Alembert's formula.