Hello,
I am struggling with the following question:
Find all the solutions of the equation
(d/dt (u(x,t))) -(d/dx (u(x,t))) +2u(x,t)=0
Hint: a change on coordinates to simplify (d/dt (u(x,t))) -(d/dx (u(x,t)))
Any help woyld be appreciated.
Hello,
I am struggling with the following question:
Find all the solutions of the equation
(d/dt (u(x,t))) -(d/dx (u(x,t))) +2u(x,t)=0
Hint: a change on coordinates to simplify (d/dt (u(x,t))) -(d/dx (u(x,t)))
Any help woyld be appreciated.
Maybe this will help you. See my last post here:
http://www.mathhelpforum.com/math-he...-182858-2.html
I think you and the OP are talking about different problems.
To the OP. If you introduce a change of variable from $\displaystyle (t,x)$ to $\displaystyle (r,s)$ then from the chain rule
$\displaystyle u_t = u_r r_t + u_s s_t$
$\displaystyle u_x = u_r r_x + u_s s_x$
so your original problem becomes
$\displaystyle u_r r_t + u_s s_t - \left( u_r r_x + u_s s_x\right) + 2u = 0$
or
$\displaystyle \left(r_t- r_x\right)u_r +\left(s_t - s_x\right)u_s + 2 u = 0$.
If you can chose r and s such that
$\displaystyle r_t - r_x = 0$ and $\displaystyle s_t-s_x = 1$, the your PDE becomes $\displaystyle u_s + 2u = 0$ - and ODE!
Here's one choice
Spoiler:
See how that goes.