MATH383: A first course in differential equationsFebruary 25, 2020

Solve the following problems (3 course points each). Present a brief motivation of your method of solution. Explicitly state any conditions that must be met for solution procedure to be valid. Sketch out a solution for yourself on scratch paper, and then neatly transcribe so the solution you present is readily legible.

No credit is awarded for statement of the final answer to a problem without presentation of the solution procedure.

1. At $t=0$ an object is placed in a room with temperature of ${20}^{\circ }$ C. The temperature of the object drops by $5^{\circ }$ C in 4 minutes and by $7^{\circ }$ C in 8 minutes . What was the temperature of the object at $t=0$?

2. Determine whether $𝑩$ is a basis set for real-valued 2 by 2 matrices

   ${\displaystyle 𝑩=\left\{\begin{array}{lll}{𝑩}_1,& {𝑩}_2,& {𝑩}_3\end{array}\right\}=\left\{\begin{array}{lll}\left(\begin{array}{ll}1& 3\\ 2& 1\end{array}\right),& \left(\begin{array}{ll}-1& 2\\ 1& 0\end{array}\right),& \left(\begin{array}{ll}0& 1\\ 0& -4\end{array}\right)\end{array}\right\}.}$

3. Let

   ${\displaystyle 𝒮=\{\left(\begin{array}{l}2s-t\\ s\\ t\\ -s\end{array}\right),s,t\in ℝ\}.}$

   1. Prove that $(𝒮,+,ℝ,\cdot )$ a subspace of $({ℝ}^4,+,ℝ,\cdot )$.

   2. Find two vectors that span $𝒮$.

4. Find and subsequently sketch the solution to the initial value problem

   ${\displaystyle y^{\text{'}\text{'}}-14y^{\text{'}}+49y=0,y\left(1\right)=2,y^{\text{'}}\left(1\right)=11.}$