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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