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[SOLVED] a^2 u_xx=u_t

karush

Well-known member
Jan 31, 2012
2,928
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
Jan 26, 2012
1,430
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
Jan 31, 2012
2,928
$${u}_{xx}=\d{^2 }{x^2 }\left(u\right)=
\left(\frac{{x}^{2}\sqrt{\frac{\pi}{t}}}{4 a^4 t^2 }
-\frac{\sqrt{\frac{\pi}{t}}}{2{a}^{2}t} \right)
\cdot e^{\frac{x^2 }{4{a}^{2}t}}$$

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