We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$. We set $a = (a_1, a_2)$ and $u_a(t) = (x_1(t), x_2(t))$. We assume that $\lambda_1 \neq 0$ and $a_1 > 0$. Determine a function $\theta_a$ such that $x_2(t) = \theta_a(x_1(t))$ for every real $t$.
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem
$$X' = AX, \quad X(0) = a$$
We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
We set $a = (a_1, a_2)$ and $u_a(t) = (x_1(t), x_2(t))$. We assume that $\lambda_1 \neq 0$ and $a_1 > 0$. Determine a function $\theta_a$ such that $x_2(t) = \theta_a(x_1(t))$ for every real $t$.