Added Nov 25, 2018.
Problem 12.2.5 (d) from Richard Haberman applied partial differential equations book, 5th edition
Solve for
with
See my HW 12, Math 322, UW Madison.
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, t], t] + 3*t*D[w[x, t], x] == w[x, t]; ic = w[x, 0] == f[x]; sol = AbsoluteTiming[TimeConstrained[Simplify[DSolve[{pde, ic}, w[x, t], {x, t}]], 60*10]];
Maple ✓
restart; pde := diff(w(x,t),t)+3*t*diff(w(x,t),x)=w(x,t); ic:=w(x,0)=f(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],w(x,t))),output='realtime'));
Hand solution
With initial conditions
Solution
Comparing (1,2) shows that
Solving (3) gives
From initial conditions at
From (4)
Substituting the above in (5) gives
Alternative solution
Using the method of characteristics, the systems of characteristic lines are (from the
PDE itself)
With initial conditions at
Equation (2) gives, after replacing
Solving for
Using (7,5) in the above gives the solution
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