Tutorial Week 3

Worksheet Questions

1. Using Reduction of Order, solve the following differential equations to find a second solution, and then state the general solution.
   1. $y''+y = 0$ where $y_1(t) = cos(t)$ is a solution.
   2. $2t^2y''+ty'-3y = 0$ where $y_1(t) = \dfrac{1}{t}$ is a solution.
2. Use the Method of Undetermined Coefficients to answer the following question.
   1. Determine a particular solution to $y''-4y'+4y = e^{2t}$
   2. Write the general solution to the above equation.
   3. What does the general solution become if $y(0) = 2$ and $y'(0) = 5$?
3. Optional: You have a garden and are beginning to see weeds. Weeds spread at a per-capita rate of 2/day, and naturally die at a rate of 0.3/day.
   1. Write a differential equation for the total number of weeds at time t, $\dfrac{dW_1}{dt}$.
   2. You decide to take action and start weeding your garden, removing 20 weeds every day. What is the new differential equation,$\dfrac{dW_2}{dt}$, for the situation?
   3. Find the solution to both differential equations if the initial number of weeds was 10.
   4. Sketch a graph showing these two solutions.
   5. Is there ever a time in either situation where your garden is weed-free? If so, find the time.

Solutions

1. 1. $y(t) = c_1sint+c_2cost$
   2. $y(t) = c_1t^{3/2}+c_2\dfrac{1}{t}$
2. 1. $y_p(t) = \dfrac{1}{2}t^2e^{2t}$
   2. $y(t) = c_1e^{2t}+c_2te^{2t}+\dfrac{1}{2}t^2e^{2t}$
   3. $y(t) = \dfrac{1}{2}e^{2 t} (t^2+2 t+4)$
3. 1. $\dfrac{dW_1(t)}{dt} = 2W_1-0.3$
   2. $\dfrac{dW_2(t)}{dt} = 2W_2-20.3$
   3. $W_1 = 0.15+9.85e^{2t}, W_2 = 10.15-0.15e^{2t}$
   4. graph
   5. In situation 1, the weeds grow exponentially and the garden is never weed-free. In situation 2, the weeds decrease exponentially and the garden becomes weed-free after t = 1.85 days.