[Step-by-Step] Consider the heat equation (partial u)/(partial t)=(partial^2 u)/(∂ x^2) Show that if u(x, t)=t^alpha φ(\xi) where \xi=x / √t and
Question:
Consider the heat equation
\[\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}\]Show that if \(u(x, t)=t^{\alpha} \phi(\xi)\) where \(\xi=x / \sqrt{t}\) and \(\alpha\) is a constant, then \(\phi(\xi)\) satisfies the ordinary differential equation
\[\alpha \phi-\frac{1}{2} \xi \phi^{\prime}=\phi^{\prime \prime}\](where \(\prime \equiv d/d\xi \) ). Show that
\[\int_{-\infty}^{\infty} u(x, t) d x=\int_{-\infty}^{\infty} t^{\alpha} \phi(\xi) d x\]is independent of \(t\) only if \(\alpha=-\frac{1}{2}\). Further, show that if \(\alpha=-\frac{1}{2}\) then
\[C-\frac{1}{2} \xi \phi=\phi^{\prime}\]where \(C\) is an arbitrary constant. From this last ordinary differential equation, and assuming \(C=0\), deduce that
\[u(x, t)=\frac{A}{\sqrt{t}} e^{-x^{2} / 4 t}\]is a solution of the heat equation (here \(A\) is an arbitrary constant).
Show that as \(t\) tends to zero from above,
\[\lim _{t \rightarrow 0+} \frac{1}{\sqrt{t}} e^{-x^{2} / 4 t}=0 \text { for } x \neq 0\]and that for all \(t>0\)
\[\int_{-\infty}^{\infty} \frac{1}{\sqrt{t}} e^{-x^{2} / 4 t} d x=B\]where \(B\) is a (finite) constant. Given that \(\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}\), find \(B\).
What physical and/or probabilistic interpretation might one give to this solution \(u(x, t)\) ?
Deliverable: Word Document 
Solution: Let X_1, X_2, ..., X_n be independent, identically
Solution: Let X and Y be random variables with joint probability
Solution: Assume that the probability of the number of breakdowns
(See Steps) Apply the slope predictor formula to find the slop
![[All Steps] Given f(x)=(x-4)/(x^2-16), find all points](/images/solutions/MC-solution-library-37636.jpg "[All Steps] Given f(x)=(x-4)/(x^2-16), find all points where f")
[All Steps] Given f(x)=(x-4)/(x^2-16), find all points where f
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