Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz5 => Topic started by: Victor Ivrii on March 09, 2018, 05:51:10 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 $t \ge 0$.
$$\left\{\begin{aligned}
&x'_1 = 1.25x_1 + 0.75x_2, &&x_1(0) = 2,\\
&x'_2= 0.75x_1 + 1.25x_2, &&x_2(0) = 1.
\end{aligned}\right.$$

a) Isolate $x_2$ in equation 1 we get
$$x_2 = \frac{4}{3}x_1'  \frac{5}{3}x_1$$
Differentiate both sides with respect to $t$ we get
$$x_2' = \frac{4}{3}x_1''  \frac{5}{3}x_1'$$
Substitute into the second equation and simplify, we get $$ x_1''  \frac{5}{2}x_1' + x_1 = 0 $$
which is a second order ODE of $x_1$.
b)
Characteristic equation is $r^2  \frac{5}{2} r + 1 = (r  \frac{1}{2})(r  2) = 0$ with roots $r_1 = \frac{1}{2}, r_2 = 2$
General solution for $x_1$ is $x_1 = c_1 e^{\frac{1}{2}t} + c_2 e^{2t}$
Plug in to $x_2 = \frac{4}{3}x_1'  \frac{5}{3}x_1$ get
$$x_2 = c_1 e^{\frac{1}{2}t} + c_2 e^{2t}$$
So, $$x_1 = c_1 e^{\frac{1}{2}t} + c_2 e^{2t}$$ $$x_2 = c_1 e^{\frac{1}{2}t} + c_2 e^{2t}$$
Plug in $x_1(0)=2, x_2(0) = 1$ to get $$2 = c_1 + c_2 $$ $$1= c_1 + c_2 $$
Solve the linear system we have
$$c_1 = \frac{3}{2}, c_2 = \frac{1}{2}$$
That is, $$x_1 = \frac{3}{2} e^{\frac{1}{2}t} \frac{1}{2} e^{2t}$$ $$x_2 = \frac{3}{2} e^{\frac{1}{2}t} \frac{1}{2} e^{2t}$$
c) See attached picture
Note that as $t \to \infty$, the graph is asymptotic to the line $x_2 = x_1$ in the third quadrant.