Toronto Math Forum
APM346-2016F => APM346--Tests => TT1 => Topic started by: Victor Ivrii on October 19, 2016, 10:24:30 PM
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Consider the first order equation:
\begin{equation}
u_t + xt u_x = - u.
\label{eq-1-1}
\end{equation}
(a) Find the characteristic curves and sketch them in the $(x,t)$ plane.
(b) Write the general solution.
(c) Solve equation (\ref{eq-1-1}) with the initial condition $u(x,0)= (x^2+1)^{-1}$.
Explain why the solution is fully determined by the initial condition.
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a) Characteristic Equation
\begin{equation} \frac{dt}{1} = \frac{dx}{xt} = \frac{du}{-u} \end{equation}
From $ \frac{dt}{1} = \frac{dx}{xt} $, $\frac{t^2}{2} + \ln c = \ln x$, thus $ x = ce^\frac{t^2}{2}$
b) General Solution
From $ \frac{dt}{1} = \frac{du}{-u}$, $-t + \ln k = \ln u$
So $u = ke^{-t} = \phi(xe^{\frac{-t^2}{2}})e^{-t} $
c) Since $u|_{t=0} = \frac{1}{1+x^2}$,
$\phi(x) = \frac{1}{1+x^2} $
Therefore, \begin{equation} u(x,t) = \frac{1}{1+x^2e^{-t^2}}e^{-t}\end{equation}
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Just to add onto Roro Sihui Yap's solution for (c), the solution is fully determined by condition at $t=0$ because all characteristics intersect the $x$ axis and do not intersect with each other.
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:D