grandes-ecoles 2014 QIID2

grandes-ecoles · France · centrale-maths1__pc Second order differential equations Reduction of a differential system to a second-order ODE

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 $$X' = AX, \quad X(0) = a$$
Show that the relation $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ holds when the matrix $A$ has a characteristic polynomial that splits over $\mathbb{R}$.

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
$$X' = AX, \quad X(0) = a$$

Show that the relation
$$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$
holds when the matrix $A$ has a characteristic polynomial that splits over $\mathbb{R}$.

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