# [SOLVED]a^2 u_xx=u_t

#### karush

##### Well-known member
Given
$${a}^{2}{u}_{xx}={u}_{t}$$
Is
$$u=\left(\pi/t\right)^{1/2}e^{{-x^2 }/{4a^2 t}}, \ \ t>0$$
A solution to the differential equation

$${u}_{xx }$$
Was kinda hard to get, the TI-NSPIRE returned a very complicated answer
and it doesn't look like the differential equation is true

#### Prove It

##### Well-known member
MHB Math Helper
Given
$${a}^{2}{u}_{xx}={u}_{t}$$
Is
$$u=\left(\pi/t\right)^{1/2}e^{{-x^2 }/{4a^2 t}}, \ \ t>0$$
A solution to the differential equation

$${u}_{xx }$$
Was kinda hard to get, the TI-NSPIRE returned a very complicated answer
and it doesn't look like the differential equation is true
Well if \displaystyle \begin{align*} u = \left( \pi\,t \right) ^{\frac{1}{2}}\,\mathrm{e}^{-\frac{x^2}{4\,a^2\,t}} \end{align*}? then what is \displaystyle \begin{align*} u_t \end{align*}? What is \displaystyle \begin{align*} u_{x\,x} \end{align*}? Is the DE true in this case?

#### karush

##### Well-known member
$${u}_{xx}=\d{^2 }{x^2 }\left(u\right)= \left(\frac{{x}^{2}\sqrt{\frac{\pi}{t}}}{4 a^4 t^2 } -\frac{\sqrt{\frac{\pi}{t}}}{2{a}^{2}t} \right) \cdot e^{\frac{x^2 }{4{a}^{2}t}}$$

This is what the TI-Nspire returned for $U_{xx}$
$u_t$ looked more complicated and was very different so assume DE is not true

I like to see how these derivatives were derived but that a ton of latex