grandes-ecoles 2013 QIV.A

grandes-ecoles · France · centrale-maths1__psi Second order differential equations Properties of special function solutions

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using the bound from question II.E.2, show that $\varphi _ { 0 } ( 3 ) < 0$. Deduce that $\varphi _ { 0 }$ has a zero $\left. \alpha _ { 0 } \in \right] 0,3 [$.
We admit that this is the first zero of $\varphi _ { 0 }$, that is, $\varphi _ { 0 }$ does not vanish on $] 0 , \alpha _ { 0 } [$.

We introduce the differential equation

$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$

Using the bound from question II.E.2, show that $\varphi _ { 0 } ( 3 ) < 0$. Deduce that $\varphi _ { 0 }$ has a zero $\left. \alpha _ { 0 } \in \right] 0,3 [$.

We admit that this is the first zero of $\varphi _ { 0 }$, that is, $\varphi _ { 0 }$ does not vanish on $] 0 , \alpha _ { 0 } [$.

Paper Questions

QI.A.1 QI.A.2 QI.B QI.C QI.D QII.A QII.B QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.D QII.E.1 QII.E.2 QII.E.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2