Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz5 => Topic started by: Victor Ivrii on March 09, 2018, 05:46:05 PM

a. Transform the given system into a single equation of second order.
b. Find $x_1$ and $x_2$ that also satisfy the given initial conditions.
c. Sketch the graph of the solution in the $(x_1,x_2)$plane for $t \ge 0$.
$$\left\{\begin{aligned}
& x'_1= 0.5x_1 + 2x_2, &&x_1(0) = 2,\\
&x'_2= 2x_1  0.5x_2, &&x_2(0) = 2
\end{aligned}\right.$$

Solving the first equation for B , we obtain $x_2 = x_1'/2+x_1/4$. Substitution into the second equation results in $$x_1''/2+x_1'/4 = 2x_1(x_1'/2+x_1'/4)/2$$.
Rearranging the terms, the single differential equation for $x_1$ is $$x_1''+x_1'+\frac{17}{4}x_1=0$$
The general soln is $$x_1(t) = e^{t/2}(c_1cos2t+c_2sin2t)$$. With $x_2$ given in terms of $x_1$, it has
$$x_2(t) = e^{t/2}(c_1cos2t+c_2sin2t)$$.
Imposing the specified initial conditions, we obtain $c_1 = 2, c_2 = 2$. Hence,
$$x_1(t) = e^{t/2}(2cos2t+2sin2t)$$.
$$x_2(t) = e^{t/2}(2cos2t+2sin2t)$$.
Attached is the graph.

You should write \cos (and so on)