Linear System Of Differential Equations - 2

Calculus Level 2

$\begin{cases} \dot{x}_1 = 3x_1 - x_2 + u \\ \dot{x}_2 = 5x_1 - x_2 \end{cases}$

Here $x_1$ , $x_2$ and $u$ are functions of time $t$ . Consider the case when $u(t) = 0$ . What happens to the solutions $x_1(t)$ and $x_2(t)$ (subject to any initial conditions except $x_1(0)=x_2(0)=0$ ) after a very long time?

Bonus: Attempt to derive a general relation between $x_1$ and $x_2$ which is independent of time for this case where $u = 0$

Also try this . There will be more follow up problems on this.

They converge to a non zero value The solutions diverge towards infinity Cannot say. Depends on the initial conditions. They converge to zero

2 solutions

Steven Chase
Sep 14, 2019

Since $u = 0$ , this is a homogeneous equation, similar to the previous problem. The eigenvalues for the system matrix are $1 \pm j$ . In other words, they are complex with positive real parts. So the solutions will oscillate, and their envelopes will grow without bound (diverge).

A Former Brilliant Member
Sep 14, 2019

Solving the given two differential equations, we get $\dfrac{d^2x_2}{dt^2}-$ $2\dfrac{dx_2}{dt}+$ $2x_2=0$ and a similar equation for $x_1$ . So $x_2$ grows exponentially with time, and so also $x_1$

Linear System Of Differential Equations - 2

2 solutions

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