# MA2051 - Ordinary Differential EquationsOld Exam 1: Solutions

Exam 1 Questions

1.
(a) The homogeneous solution is and a particular solution is . (The method of undetermined coefficients is perhaps the simplest way to find here.) The general solution is where C is an arbitrary constant.

(b) If you divide by x to put the equation in "standard" form, you find the integrating factor is . So the original equation is really and you just integrate to find the general solution , where C is an arbitrary constant.

2.
(a) The homogeneous solution is . (Use the characteristic equation.)

(b) Use the method of undetermined coefficients to obtain

(c) The general solution is where C is an arbitrary constant. The initial data gives you .

(d) There is no steady state solution (i.e. no constant solution). The exponential part of the solution is transient. The remaining part of the general solution is the limiting periodic solution.

3.
(a) The amount of TCE in the lake at time t is grams. So the change in this amount would be grams.

(b) No TCE is flowing into the lake, so the rate in is zero. The rate at which TCE flows out of the lake is grams per day.

(c) The balance law gives you

Divide by and take the limit as to obtain the model:

(d) If the rate of inflow is instead of zero, the model becomes

4.
The Euler approximation is . Take and solve this equation for to obtain the approximate tripling time years.
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