Study of a parameter in a system of ODEs

Question

Consider $a\in\mathbb{R}$ and a system $2\times 2$ of ODEs $$\begin{cases} x'(t)=x(t)+ay(t) \\ y'(t)=a y(t) \end{cases} . $$ I should find the values of $a$ s.t. there exists only a solution not identically null $(x(\cdot),y(\cdot))$ such that $(x(\cdot),y(\cdot))\rightarrow (0,0)$ for $t\rightarrow \infty$. Any idea?

Answer 1

$$y'=ay \quad\to\quad y=c_1e^{at}$$ I suppose that you know how to solve the separable ODE $y'=ay$ . $$x'=x+ay=x+ac_1e^{at}$$ $$x'-x=ac_1e^{at} \quad\to\quad x=c_2e^t+c_1\frac{a}{a-1}e^{at}$$ I suppose that you know how to solve the first order non-homogeneous ODE $x'-x=ac_1e^{at}$ .

Condition $x(t\to\infty)\to 0$ implies $c_2=0$ and $a<0$

As well $x(t\to\infty)\to 0$ with $a<0$ .

With given initial condition $x(t)=x_0$ and $y(0)=y_0$

$y_0=c_1$ and $x_0=c_1\frac{a}{a-1}$ imply $x_0=y_0\frac{a}{a-1} \quad\to\quad a=\frac{x_0}{x_0-y_0}$

The condition to have only one solution not identically null and $x(\infty)=0$ and $y(\infty)=0$ is $a=\frac{x_0}{x_0-y_0}<0$ .