1
$\begingroup$

Show by direct substitution that

$ P(x,t) = \frac{1}{\sqrt{4 \pi D t}} \exp \left( - \frac{x^2}{4Dt} \right) $

is a solution to the equation

$ \frac{\partial P}{\partial t} = D \frac{\partial^2 P}{\partial x^2}$

(Please show the full working and reasoning including any annotations that would help a beginner to understand.)

What does it mean by direct substitution? Direct substitution of what into what? What common mathematical techniques am I expected to use here?

  • 4
    Show by direct substitution that $x={\color{red}2}$ is a solution to the equation $x^2=x+2$: ${\color{red}2}^2={\color{red}2}+2$.2011-11-21

1 Answers 1

2

You know that $P$ is a certain function of $x$ and $t$.

So you can find $\dfrac{\partial^2 P}{\partial x^2}$ and $\dfrac{\partial P}{\partial t}$.

Take whatever you got when you found $\dfrac{\partial P}{\partial t}$ and put it in place of $\dfrac{\partial P}{\partial t}$ in the differential equation $\dfrac{\partial P}{\partial t} = D\dfrac{\partial^2 P}{\partial x^2}$. That is "direct substitution". Likewise, take whatever you got when you found $\dfrac{\partial^2 P}{\partial x^2}$ and put it in place of $\dfrac{\partial^2 P}{\partial x^2}$ in the differential equation. That is "direct substitution". Then check that the thing to the left of "$=$" actually is equal to the thing to the right. If they are equal, then the given function $P(x,t)$ is a solution of the differential equation; otherwise it's not.