Math 235, Spring 2015: Comments on Exam 1, Problem 1

In this problem, students were first asked to verify that y(t) = \frac{1}{5-t} was a solution to the differential equation
\displaystyle \frac{dy}{dt} = y^2
and to plot that function.

  • Remember when verifying that a function is a solution to a differential equation that one cannot assume that the equation is satisfied. The easiest way to proceed is to consider the left side \frac{dy}{dt} independently from the right side y^2, and then to compare the two.
  • When sketching a plot of a function, it is only necessary to plot a few “key” or “relevant” points. For the function y(t) = \frac{1}{5-t}, it is important to note the vertical intercept at y=1/5 and the asymptote at y=5. No other points need to be indicated.

    It is also important to label axes.

  • Remember not to abuse notation! Math symbols have specific meaning; be sure that you intend the meaning they impart!

For the second part of the problem, students were asked to consider another solution \tilde y(t) satisfying the same differential equation, but with initial value \tilde y(0)=2.

I was hoping that students would invoke the fundamental theorem of ODEs. This requires considering the function f(y) = y^2 and stating that both f and f^\prime(y) = 2y are continuous for all values of $y$.

  • Note: These properties of f are independent of the differential equation; they are only properties about the squaring function.

Once we know that FTODE can be applied, then we know that the solution \tilde y and y cannot cross. Therefore, since \tilde y(0)> y(0), we know that \tilde y(t) > y(t) so long as the solution exists.
In particular, it must be the case that \tilde y(t) blows up at some time, no later than t=5.

  • It is important to justify that FTODE is applicable in this setting.
  • It is not enough to describe how \tilde y behaves; one must explain the reasoning that leads to this description (by invoking FTODE).
  • There were several instances of the phrase “grows exponentially” being used improperly. We discussed in class what these words mean in a math setting.

Several students found the exact formula for \tilde y, which was not too hard. Even if one has the exact formula, one still needs to precisely describe the behavior.

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