grandes-ecoles 2014 QIIC2

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 $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
In this question, $a = (2,1)$ and $b = (1,2)$.
For each of the following cases, illustrate on the same figure the graphs of the functions $\theta_a$, $\theta_b$ and $\theta_{a+b}$, as well as the parallelograms with vertices $(0,0)$, $u_a(t)$, $u_b(t)$ and $u_a(t) + u_b(t)$ for $t = 0$ and a strictly positive value of $t$.
a) $\lambda_1 = 1$ and $\lambda_2 = 2$.
b) $\lambda_1 = 1$ and $\lambda_2 = -2$.
c) $\lambda_1 = 1$ and $\lambda_2 = -1$.

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 $A = \operatorname{diag}(\lambda_1, \lambda_2)$.

In this question, $a = (2,1)$ and $b = (1,2)$.

For each of the following cases, illustrate on the same figure the graphs of the functions $\theta_a$, $\theta_b$ and $\theta_{a+b}$, as well as the parallelograms with vertices $(0,0)$, $u_a(t)$, $u_b(t)$ and $u_a(t) + u_b(t)$ for $t = 0$ and a strictly positive value of $t$.

a) $\lambda_1 = 1$ and $\lambda_2 = 2$.

b) $\lambda_1 = 1$ and $\lambda_2 = -2$.

c) $\lambda_1 = 1$ and $\lambda_2 = -1$.

Paper Questions

QIA QIB1 QIB2 QIC1 QIC2 QIC3 QIIA QIIB1 QIIB2 QIIB3 QIIC1 QIIC2 QIID1 QIID2 QIID3 QIIIA QIIIB1 QIIIB2 QIIIB3 QIIIB4 QIIIC QIVA1 QIVA2 QIVA3 QIVB QIVC