MAT244-2018S > Quiz-5

Q5--T0501, T5101

(1/1)

Victor Ivrii:
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= x_1 - 2x_2, &&x_1(0) = -1,\\ &x'_2= 3x_1 - 4x_2, &&x_2(0) = 2. \end{aligned}\right.

Junya Zhang:
a) Isolate $x_2$ in equation 1 we get
$$x_2 = \frac{1}{2}x_1 - \frac{1}{2}x_1'$$
Differentiate both sides with respect to $t$ we get
$$x_2' = \frac{1}{2}x_1' - \frac{1}{2}x_1''$$
Substitute into the second equation and simplify, we get $$x_1'' + 3 x_1' + 2x_1 = 0$$
which is a second order ODE of $x_1$.

b)
Characteristic equation is $r^2 +3 r + 2 = (r + 2)(r + 1) = 0$ with roots $r_1 = -2, r_2 = -1$
General solution for $x_1$ is $x_1 = c_1 e^{-2t} + c_2 e^{-t}$
Plug in to $x_2 = \frac{1}{2}x_1 - \frac{1}{2}x_1'$ get
$$x_2 = \frac{3}{2}c_1 e^{-2t} + c_2 e^{-t}$$
So, $$x_1 = c_1 e^{-2t} + c_2 e^{-t}$$ $$x_2 = \frac{3}{2}c_1 e^{-2t} + c_2 e^{-t}$$
Plug in $x_1(0)=-1, x_2(0) = 2$ to get $$-1 = c_1 + c_2$$  $$2= \frac{3}{2}c_1 + c_2$$
Solve the linear system we have
$$c_1 = 6, c_2 = -7$$
That is, $$x_1 = 6 e^{-2t} -7 e^{-t}$$ $$x_2 = 9 e^{-2t} -7 e^{-t}$$

c) See attached image
Note that as $t\to \infty$, the graph approaches the origin in the third quadrant tangent to the line $x_1 = x_2$.

Darren Zhang:
Also, we can Sketch the graph.
(c) Attached is the graph
Edit: Glad to see Junya add the graph afterwards.