Solve the PDE by factoring

Question

How do you solve this PDE? $$u_{tt}-5u_{xt}-36u_{xx}=0$$

I have factored it to:

$$\left(\dfrac{\partial}{\partial t} - 9\dfrac{\partial}{\partial x}\right) \left(\dfrac{\partial}{\partial t} + 4\dfrac{\partial}{\partial x}\right)u=0$$

Then I separate the two to get:

$\left(\dfrac{\partial}{\partial t} - 9\dfrac{\partial}{\partial x}\right)u_1=0$

$\left(\dfrac{\partial}{\partial t} + 4\dfrac{\partial}{\partial x}\right)u_2=0$

I think I am supposed to integrate these after? (Please correct me if I am wrong). If so, I am not sure how to integrate them.

Thank you in advance.

Answer 1

Those two are known as (linear) transport equations (with constant coefficients) https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation

the solution to these is $u_1(x,t) = u_1^0(x+9t)$ for the first and an analogous form for the other one. Now who is this $u^0_1$ I mentioned? You should be getting that from the starting data of you original PDE, without these boundary values the PDE doesn't infact make any sense.

Answer 2

Note that both PDEs in $u_{1}$ and $u_{2}$ are linear transport equations with constant coefficients. Thus, the first equation has the solution $$ u_{1}(t,x)=f(x+9t) $$ and analogously for $u_{2}$ $$ u_{2}(t,x)=g(x-4t) $$ when $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are sufficiently smooth functions and would usually represent initial data.

Answer 3

You factored incorrectly, it should be $$ (\partial_t+9\partial_x)(\partial_t-4\partial_x)u=0 $$ This factorizations suggests the change of variables $$ \xi=x-9t\\ \eta=x+4t $$ since under this change of variables, your pde becomes $$ u_{\xi\eta}=0 $$ (check this), now integrating twice yields $$ u(\xi,\eta)=f(\xi)+g(\eta)\implies u(t,x)=f(x-9t)+g(x+4t) $$ for arbitrary functions $f$ and $g$.