Linear System of Differential Equations

Calculus Level 2

$\dot{x}_1 = -x_1 - x_2$ $\dot{x}_2 = x_1 - x_2$

Here $x_1$ and $x_2$ are functions of time $t$ . What happens to the solutions $x_1(t)$ and $x_2(t)$ (subject to any initial conditions) after a very long time?

Bonus: Attempt to derive a general relation between $x_1$ and $x_2$ which is independent of time.

They converge to a non zero value They converge to zero Cannot Say The solutions diverge towards infinity

2 solutions

Steven Chase
Sep 5, 2019

The system has the form:

$\dot{x} = A x$

The solutions therefore take the form:

$x = c_1 e^{\lambda_1 t} \vec{v}_1 + c_2 e^{\lambda_2 t} \vec{v}_2$

In the above equation, the $\lambda$ values are the eigenvalues of the matrix $A$ , and the $\vec{v}$ vectors are the eigenvectors.

As it turns out, the eigenvalues of the matrix are complex, with negative real parts. So the solutions are oscillatory, but their envelopes tend toward zero.

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 $x_2$ decays exponentially with time and so also $x_1$ .

Linear System of Differential Equations

2 solutions

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