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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1997 • Question 8

Question

Math 6410 • ODE Fall 1997 • Question 8

Assume \(f(x,t)=0\). Show by example that if \[ \varphi (x)= \begin {cases} \alpha , & \text {if }x<0,\\ \beta , & \text {if }x\ge 0. \end {cases} \] where \(\alpha <\beta \) are constants, then the IVP (3) may have more that one weak solution. (Check that your examples are weak solutions.)

Raw LaTeX

Assume \(f(x,t)=0\). Show by example that if
\[
  \varphi(x)=
  \begin{cases}
    \alpha, & \text{if }x<0,\\
    \beta, & \text{if }x\ge0.
  \end{cases}
\]
where \(\alpha<\beta\) are constants, then the IVP (3) may have more that one weak
solution. (Check that your examples are weak solutions.)

Topic Tags

Distributions and Weak Solutions Poincare Maps and Periodic Solutions Sobolev Spaces and Energy Estimates Variation of Constants Formula